# PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

Size: px
Start display at page:

Download "PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures."

Transcription

1 PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the distance around the polygon, i.e. the sum of the lengths of its sides. Examples Answer the following. Remember to include units. 1. Find the perimeter of an equilateral triangle with side length 7 m.. Find the perimeter of a square with side length 5 in. 3. Find the perimeter of a regular hexagon with side length 10 m. 4. Find the perimeter of a rectangle with length 8 cm and width 5 cm. 5. If the perimeter of a regular pentagon is 75 in, find the length of one side. 6 If a tennis court has a perimeter of 10 ft and a length of 78 ft, find its width. 7 Find the perimeter of the following swimming pool. (Assume that all intersecting sides in the diagram are perpendicular to each other.) 10 m 13 m Pool 3 m 9 m The solutions to the above examples can be found below.

2 Solutions to Examples: 1. Find the perimeter of an equilateral triangle with side length 7 m. 7 m P = = 3(7) = 1 The perimeter of the equilateral triangle is 1 m.. Find the perimeter of a square with side length 5 in. P = = 4(5) = 0 5 in The perimeter of the square is 0 in. 3. Find the perimeter of a regular hexagon with side length 10 m. 10 m P = = 6(10) = 60 The perimeter of the regular hexagon is 60 m. 4. Find the perimeter of a rectangle with length 8 cm and width 5 cm. 5 cm P = = (8) + (5) = 36 8 cm The perimeter of the rectangle is 36 cm. Note: The perimeter of a rectangle with length l and width w can be represented by the formula: P = l+ w.

3 5. If the perimeter of a regular pentagon is 75 in, find the length of one side. Let x = the length of a side of the regular pentagon Since the perimeter of the pentagon is 75 inches, we can say that x+ x+ x+ x+ x= 75, or equivalently 5x = 75. x in Next, we solve for x by dividing both sides by 5: x = 15 The length of one side of the regular pentagon is 15 inches. 6 If a tennis court has a perimeter of 10 ft and a length of 78 ft, find its width. Let x = the width of the tennis court x ft 78 ft We know that P = 10. Using the formula l+ w= P, (78) + x = 10. Solving for x: x = 10 x = 54 x = 7 Therefore, the width of the tennis court is 7 ft. 7. Find the perimeter of the following swimming pool. (Assume that all intersecting sides in the diagram are perpendicular to each other.) 13 m 3 m 10 m Pool 9 m

4 Let x and y represent the unlabeled sides of the pool, as shown in the diagram below. 13 m 3 m 10 m Pool y m x m 9 m The horizontal distance across the pool is 13 m (as seen along the top edge of the pool). Therefore, 9+ x = 13, so x = 4. The vertical distance across the pool is 10 m (as seen along the left edge of the pool). Therefore, 3+ y = 10, so y = 7. Now, adding the side lengths to find the perimeter of the figure: P = = 46 The perimeter of the pool is 46 m.

5 Circumference The circumference of a circle is the distance around the circle. If a circle is compared to a many-sided polygon, the circumference of a circle can be likened to the perimeter of a polygon. Let us first begin with an exploration regarding circumference. Consider the table below, which contains a listing of various circular objects (lid, soda can, drinking glass, jar, etc.) In each of these three-dimensional objects, a plane containing a circle has been isolated, and the diameter and circumference of that circle has been listed. Measurements have been rounded to the nearest tenth of a centimeter. OBJECT DIAMETER CIRCUMFERENCE CIRCUMFERENCE DIAMETER Lid (outer rim) 6.1 cm 19.3 cm Soda Can (top rim) 5.5 cm 17. cm Drinking Glass (top rim) 8. cm 5.7 cm Jar (top rim) 4.7 cm 14.9 cm Mixing Bowl (top rim) 1.6 cm 67.8 cm Bucket (top rim) 31.0 cm 97.5 cm Exercises 1. Use a calculator to compute the ratio Circumference Diameter for each object in the last column of the table above. Round this ratio to the nearest thousandth.. Find three other circular objects. Add them to the table above, measure the circumference and diameter for each, and then compute the ratio Circumference Diameter for each object. (Note: To measure the circumference of an object, you may want to wrap a piece of string around the object, and then measure the string.) 3. Analyze the last column of numbers in the table. What do you notice? A completed chart can be found below.

6 OBJECT DIAMETER CIRCUMFERENCE CIRCUMFERENCE DIAMETER Lid (outer rim) 6.1 cm 19.3 cm Soda Can (top rim) 5.5 cm 17. cm 3.17 Drinking Glass (top rim) 8. cm 5.7 cm Jar (top rim) 4.7 cm 14.9 cm Mixing Bowl (top rim) 1.6 cm 67.8 cm Bucket (top rim) 31.0 cm 97.5 cm Notice that the numbers in the last column are very close to each other in their numerical value. Ancient mathematicians noticed that the ratio Circumference Diameter always seemed to give the same number, and it became a challenge to try to determine the value of this ratio. Since we rounded to the nearest tenth (and measurement is not completely accurate due to human error and/or measuring devices), the values in our table are not exactly the same. If we were to be able to measure perfectly, we would find that the ratio Circumference Diameter is always equal to the number π (pronounced pi ). π is an irrational number, but its value is shown below to 50 decimal places: π Note: For more information on irrational numbers, see the Irrational Numbers tutorial in the Additional Materials section. Since the ratio Circumference Diameter is equal to π, we can write the following equation: C d = π (Where C represents circumference and d represents diameter.) Multiplying both sides of the equation by d, we obtain the following equation: C = π d Since the diameter d of a circle is twice the radius r, i.e. d = r, we can also write: C = π d = π( r) = πr Circumference The circumference of a circle, denoted by C, is the distance around the circle. The circumference is given by either of the following formulas: C = π d, or equivalently, C = π r, where d and r represent the diameter and radius of the circle, respectively.

7 Examples Find the circumference of the following circles. First, write the circumference of each circle in terms of π. Then use the approximation π 3.14 and compute the circumference of each circle to the nearest hundredth. (Be sure to include units in each answer.) 1. 5 ft. 1 cm Solutions: 1. The radius of the circle is 5 ft ( 5) C π r π ( 5) 10π r = so we use the equation C = π r. = = = Exact answer: C = 10π ft Using 3.14 for π : C = 10π Approximate answer: C 31.4 ft ( ). The diameter of the circle is 1 cm ( 1) C π d π ( 1) 1π d = so we use the equation C = π d. = = = Exact answer: C = 1π cm Using 3.14 for π : C = 1π Approximate answer: C cm ( )

8 Area of a Circle The formula for the area of a circle can be found below. Area of a Circle The area of a circle with radius r is given by the formula A= π r Examples Find the area of the following circles. First, write the area of each circle in terms of π. Then use the approximation π 3.14 and compute the area of each circle to the nearest hundredth. (Be sure to include units in each answer.) ft 10 cm Solutions: 1. A = πr = π 7 = 49π ft. Exact answer: Using 3.14 for π : A = 49π Approximate answer: ( ) A = 49π ft A ft. The diameter of the circle is 10 cm ( d = 10) so r = 5 cm. A = πr = π 5 = 5π cm. Exact answer: Using 3.14 for π : A = 5π Approximate answer: ( ) A = 5π cm A 78.5 cm

9 Special Right Triangles (Needed for upcoming sections involving area) There are two special right triangles which are useful to us as we study the area of polygons. These triangles are named by the measures of their angles, and are known as 45 o -45 o -90 o triangles and 30 o -60 o -90 o triangles. A diagram of each triangle is shown below: leg 45 o hypotenuse longer leg 30 o hypotenuse leg 45 o 60 o shorter leg Tutorial: For a more detailed exploration of this section along with additional examples and exercises, see the tutorial entitled Special Right Triangles in the Additional Material section. The theorems relating to special right triangles can be found below, along with examples of each. Theorem: In a 45 o -45 o -90 o triangle, the legs are congruent, and the length of the hypotenuse is times the length of either leg. Examples Find x and y by using the theorem above. Write answers in simplest radical form. 1. Solution: y The legs of the triangle are congruent, so x = 7. The x hypotenuse is times the length of either leg, so 45 o y = Solution: 13 x The hypotenuse is times the length of either leg, so the length of the hypotenuse is x. We are given that y 45 o

10 the length of the hypotenuse is 13, so x = 13, and we obtain x = 13. Since the legs of the triangle are congruent, x= y and y = Solution: 45 o 7 x The hypotenuse is times the length of either leg, so the length of the hypotenuse is x. We are given that the length of the hypotenuse is 7, so x = 7, and we y 7 obtain x =. Rationalizing the denominator, x = =. Since the legs of the triangle are congruent, x= y and y =. Note: For more information on rationalizing denominators, see the Rationalizing Denominators tutorial in the Additional Materials section. Theorem: In a 30 o -60 o -90 o triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg. Examples Find x and y by using the theorem above. Write answers in simplest radical form. 1. Solution: The length of the shorter leg is 6. Since the length of 30 o x y the hypotenuse is twice the length of the shorter leg, x = 6 = 1. The length of the longer leg is 3 times 60 o 6 the length of the shorter leg, so y = Solution: y The length of the shorter leg is x. Since the length of the 8 3 longer leg is 3 times the length of the shorter leg, the 60 length of the longer leg is x 3. We are given that the o x length of the longer leg is 8 3, so x 3 = 8 3, and therefore x = 8. The length of the hypotenuse is twice the length of the shorter leg, so y = x= 8= 16.

11 3. Solution: 9 The length of the shorter leg is y. Since the length of the y hypotenuse is twice the length of the shorter leg, 30 o 9 9= y, so y = = 4.5. Since the length of the longer x leg is 3 times the length of the shorter leg, x= y 3 = 3 = o y Solution: x The length of the shorter leg is x. Since the length of the 30 o 1 longer leg is 3 times the length of the shorter leg, the length of the longer leg is x 3. We are given that the length of the longer leg is 1, so x 3 = 1. Solving for x and rationalizing the denominator, we obtain x = = = = 4 3. The length of the hypotenuse is twice the length of the shorter leg, so y = x= 4 3 = 8 3. Area of a Parallelogram Before we explore how to find the area of a parallelogram, let us first review the definition of a parallelogram: Parallelogram A parallelogram is a quadrilateral with two pairs of parallel sides. The following quadrilaterals are examples of parallelograms: The second, third, and fourth diagrams above represent a square, rectangle, and rhombus, respectively. All of these quadrilaterals can still be classified as parallelograms, since they each have two pairs of parallel sides.

12 The area of a figure is represented by the number of square units which can be contained inside of it. Area of a Rectangle Let us first examine the following rectangle and determine its area: 3 in 5 in If we divide the rectangle into one-inch squares, we can quickly see that the area of the rectangle is 15 square inches, as shown below. 3 in 5 in We can see from this example that in order to find the area of a rectangle, we can just multiply the base times the height. In the previous example, A = 5(3) = 15 in. Area of a Rectangle The area of a rectangle with base b and height h is given by the formula A= bh

13 Area of a Square A square is simply a special case of a rectangle. The formula for the area of a rectangle can be used for squares, but because it is a special case, there is a special formula for the area of the square as well. Suppose that a square has side s. This represents both the base and the height of the square. Since the formula for the area of a rectangle is A = bh, we can say that A = s s= s. Area of a Square The area of a square of side s is given by the formula A = s. Consider a square with side length 4 cm. The area of the square is16 cm, as illustrated by the following diagrams and equations. A = s = = cm We also could use the formula A= bh: A = 4(4) = 16 4 cm The area of the square is 16 cm. This is further illustrated below by dividing the diagram into one-inch squares. 4 cm 4 cm

14 Area of a Parallelogram Suppose that we want to find the area of the following parallelogram. 5 in 3 in 4 in We want to divide the parallelogram into square units, but the sides of the parallelogram are not perpendicular to each other. Look at the triangle which is shaded in the figure below. 5 in 3 in 4 in Imagine slicing the triangle off of the parallelogram above, and translating it to the other side of the parallelogram, as shown below. This forms a rectangle which has the same area as the original parallelogram. 5 in 4 in 3 in To find the area of the rectangle, we need only to multiply the base and the height. A = 5(3) = 15 in Since the area of the original parallelogram is the same as the area of the rectangle, the area of the parallelogram is also 15 in.

15 Notice that the length of the 4 cm side of the parallelogram was extraneous in terms of finding its area; all that was needed was the base and the height. A common error in finding the area of the parallelogram above is to multiply 5 times 4; the reason this is incorrect is that the answer (0) represents slanted units, not square units (where each unit is a rhombus). A diagram is shown below to illustrate this concept. INCORRECT method of computing area: 4 in An answer of 0 square units for the area (multiplying 5 times 4) is incorrect. Although 0 units of area can be seen in the diagram at the left, they are not square units and are not a standard means of measuring area. 5 in The proper formula for the area of a parallelogram can be found below. Area of a Parallelogram The area of a parallelogram with base b and height h is given by the formula A= bh Examples Find the area of each of the following parallelograms. Be sure to include units. (Note: Some figures may not be drawn to scale.) 1. 4 m 7 m. 3 yd 3 yd

16 3. 6 in 5 in 9 in 4. 8 cm 10 cm 60 o 5. 7 ft 1 ft 45 o 6. 7 ft 1 ft 45 o Solutions: 1. Since the base of the parallelogram is 7 m and the height is 4 m, A= bh= 7(4) = 8 m.. Since the base of the parallelogram is 3 yd and the height is 3 yd, A= bh= 3(3) = 3 = 9 yd. 3. Since the base of the parallelogram is 9 in and the height is 5 in, A= bh= 9(5) = 45 in.

17 4. We first need to find the height of the parallelogram. When an altitude is drawn, it forms a 30 o -60 o -90 o triangle, as shown below. 10 cm 8 cm h 60 o x The length of the shorter leg is x. Since the length of the hypotenuse is twice 8 the length of the shorter leg, 8= x, so x = = 4. Since the length of the longer leg is 3 times the length of the shorter leg, h= x 3 = 4 3. Since the base is 10 cm and the height is 4 3 cm, A= bh= = 40 3 cm. ( ) 5. We first need to find the height of the parallelogram. When an altitude is drawn, it forms a 45 o -45 o -90 o triangle, as shown below. 1 ft 7 ft h 45 o The hypotenuse is times the length of either leg, so the length of the hypotenuse is h. We are given that the length of the hypotenuse is 7, so h = 7, and we obtain h = 7 ft. Since the base of the parallelogram is 1 ft and the height is 7 ft, A= bh= 1(7) = 84 ft. 6 We first need to find the height of the parallelogram. When an altitude is drawn, it forms a 45 o -45 o -90 o triangle, as shown below. 1 ft 7 ft h 45 o

18 The hypotenuse is times the length of either leg, so the length of the hypotenuse is h. We are given that the length of the hypotenuse is 7, so 7 h = 7, and we obtain h =. Rationalizing the denominator, h = = 7 7. Since the base of the parallelogram is 1 ft and the height is 7 ft, 7 84 A= bh= 1 = = 4 ft. ( ) Area of a Triangle Suppose that we want to find the area of the following triangle. 6 cm 10 cm First, we will mark the midpoint of one of the sides (other than the base) as the center of rotation, as shown below. 6 cm Center of rotation 10 cm We now rotate the triangle 180 o around the center of rotation. The original triangle and its image are shown below. (The image is shown as a shaded region.)

19 6 cm 10 cm Notice that the two triangles together form a parallelogram. The base of the parallelogram is 10 cm and its height is 6 cm. To find the area of the parallelogram, we multiply the base and the height, so the parallelogram has an area of 60 cm. The original triangle comprises half of the parallelogram, so the area of the original = cm. 1 triangle is ( 60) 30 The above process could be repeated for any triangle. We conclude that the area of a triangle is equal to one-half of the area of a parallelogram. Since the formula for the area 1 of a parallelogram is A= bh, the formula for the area of a triangle is A= bh. Area of a Triangle The area of a triangle with base b and height h is given by the formula: 1 A= bh. Examples Find the area of each of the following triangles. Be sure to include units. (Note: Some figures may not be drawn to scale.) 1. 4 in 7 in

20 . Find the area of the shaded triangle. 8 cm 9 cm ft 60 o 19 ft Solutions: 1. Since the base of the triangle is 7 in and the height is 4 in, A= 1 bh= 1(7)(4) = 14 in.. Since the base of the triangle is 9 cm and the height is 8 cm, A= 1 bh= 1(9)(8) = 36 cm. 3. We first need to find the base and the height of the triangle. Let us label the diagram with x and h as shown below: 14 ft h 60 o x 19 ft Notice that the altitude of the original creates a 30 o -60 o -90 o triangle with hypotenuse 14 ft and short leg x. Since the length of the hypotenuse is twice 14 the length of the shorter leg, 14 = x, so x = =. Since the length of the 7 longer leg is 3 times the length of the shorter leg, h= x 3 = 7 3.

21 The base of the original triangle is 6 ft (since 7+ 19= 6). The height of the triangle is 7 3 ft. Therefore, A= 1 bh= 1(6)(7 3) = 91 3 ft. Area of a Trapezoid Let us first review the definition of a trapezoid. Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are known as the bases and the other two sides are known as legs. Suppose that we want to find the area of the following trapezoid. 7 cm 6 cm 18 cm First, we will mark the midpoint of one of the legs as the center of rotation, as shown below. 7 cm 6 cm Center of rotation 18 cm We now rotate the trapezoid 180 o around the center of rotation. The original trapezoid and its image are shown below. (The image is shown as a shaded region.)

22 7 cm 18 cm 6 cm 18 cm 7 cm Notice that the two trapezoids together form a parallelogram. The base of the parallelogram is 5 cm (since = 5 ) and its height is 6 cm. To find the area of the parallelogram, we multiply the base and the height. 5 6 = 150, so the parallelogram has an area of 150 cm. The original trapezoid comprises half of the parallelogram, so the area of the original = 75 cm. trapezoid is ( ) The above process could be repeated for any trapezoid, as shown below in general form. In the diagram below, the bases of the trapezoid are denoted as b 1 and b, and the height is denoted by h. b 1 h b We again rotate the trapezoid 180 o around the midpoint of one of the legs; the two trapezoids together form a parallelogram. b 1 b h b b 1

23 The area of the trapezoid is one-half the area of the parallelogram formed by the two b + b and has height h. Since trapezoids. The parallelogram has a base represented by ( ) 1 the formula for the area of a parallelogram is A= bh, the area of the parallelogram is b + b h. The area of the trapezoid is one-half the area of the parallelogram; therefore ( ) 1 1 the area of the trapezoid can be represented by the formula ( ) A = b + b h. 1 Area of a Trapezoid The area of a trapezoid with bases b 1 and b and height h is given by the formula: 1 b1+ b A= ( b1+ b) h= h. One way to remember this formula is A = The average of the bases (The height). ( ) Example Find the area of the following trapezoid. Be sure to include units. 5 in 3 in 13 in Solution: The bases of the trapezoid are 5 in and 13 in, and the height is 3 in A= b + b h= = 18 3 = 7 in. ( ) ( )( ) ( )( ) 1 Suppose that we want to use the equivalent formula b1+ b A= ( The average of the bases ) (The height) = h b1+ b We can first find the average of the bases, = = 9.

24 ( The average of the bases ) (The height)= 9(3) = 7 in A =. Area of a Regular Polygon If a polygon is both equilateral (all sides are congruent) and equiangular (all angles are congruent), it is called a regular polygon. Some examples of regular polygons can be found below. Equilateral Triangle Square Regular Pentagon Regular Hexagon We will now define some other terms which relate to regular polygons. A circle can be circumscribed around any regular polygon, and the radius of the regular polygon is the same as the radius of its circumscribed circle. The radius of the regular polygon can also be defined as the distance from the center of the regular polygon to one of its vertices. (Note: The center of a regular polygon is the same as the center of its circumscribed circle.) Examples are shown below, with a radius labeled r in each diagram. (Assume that all polygons in this section are regular polygons; congruence marks have been omitted to keep the diagrams from being too cluttered.) r r r r A circle can be inscribed within any regular polygon, and the apothem of the regular polygon is the same as the radius of its inscribed circle. The apothem of the regular polygon can also be defined as the distance from the center of the regular polygon to the midpoint of one of its sides. Examples are shown below, with an apothem labeled a in each diagram.

25 a a a a A central angle of a regular polygon has the following properties: 1. Its vertex is at the center of the regular polygon.. Its sides are two consecutive radii of the regular polygon. Examples are shown below, with a central angle shown in each diagram. In order to find the area of a regular polygon, we must first learn to find the measure of one of its central angles. Suppose that we want to find the measure of a central angle of a regular hexagon. In the diagram below, all of the central angles of a regular hexagon are drawn. A regular hexagon has six central angles which are all congruent. Since the degree measure of a complete rotation is 360 o, the measure of one central angle is =.

26 We repeat the same process for a regular pentagon below. A regular pentagon has five central angles which are all congruent. Since the degree measure of a complete rotation is 360 o, the measure of one central angle is =. The Measure of a Central Angle of a Regular Polygon If a regular polygon has n sides, then the measure of one of its central angles is 360. n Exercises Answer the following: 1. Sketch each of the following regular polygons. Then draw a central angle for each and find its measure. a) Equilateral Triangle b) Square c) Regular Pentagon d) Regular Hexagon e) Regular Octagon. Find the measure of a central angle for the following regular polygons. a) Regular Decagon b) Regular Dodecagon (a dodecagon has 1 sides) c) Regular 36-gon d) Regular 10-gon

27 Suppose that we want to find the area of the following regular hexagon. 10 cm First, we draw all radii of the regular hexagon to divide it into six congruent triangles. (We will use this approach to discover a general formula for the area of any regular polygon.) 10 cm Let us examine the properties of one of the triangles above. Since a regular hexagon has 6 sides, the measure of one of its central angles is = 60. If an apothem is drawn, we can see that it forms a 30 o -60 o -90 o triangle, as shown in the enlarged view of the triangle below. Enlarged View 60 o a 30 o 10 cm 10 cm 60 o We want to find the length of the apothem. The shorter leg of the 30 o -60 o -90 o triangle has length 5 cm, since the apothem intersects the side at its midpoint. The length of the longer leg, a, is 3 times the length of the shorter leg, so a = 5 3 cm.

28 We now rearrange the six individual triangles to form a parallelogram. (The figure is again enlarged to show detail.) a = 5 3 cm 10 cm 10 cm 10 cm a 10 cm 10 cm 10 cm Figure enlarged to show detail. Notice that the parallelogram has a base of 30 cm and a height of 5 3 cm. To find the A= bh= = cm. Therefore, area of a parallelogram, we use the formula ( ) the regular hexagon with side length 10 cm has area cm. We now want to use these results to find a general formula for the area of a regular polygon. Notice that the base of the parallelogram is half of the perimeter of the regular hexagon. (The perimeter of the regular hexagon is 60 cm, and the base of the parallelogram is 30 cm.) If we denote the base of the parallelogram as b and the perimeter P of the regular polygon as P, then b =. We can see from the diagram above that the height, h, of the parallelogram is the same as the apothem, a, of the regular hexagon. P Using the formula A= bh and substituting in b = and h= a, we obtain the equation: P 1 A= bh= a= ap Side Note: If the original regular polygon had an odd number of sides (let s say a regular pentagon, for example), we would have obtained a trapezoid instead of a parallelogram when we pieced the triangles together. This diagram could still be formed into a parallelogram (more specifically, a rectangle), by slicing off one of the ends and repositioning it as shown below. a P

29 A different approach: Another method of finding the area of the regular hexagon is to find the area of one of the triangles and then multiply by 6 to find the total area (since there are six triangles formed by all of the central angles). In this case, bh 10( 5 3) The area of one triangle = = = 5 3 cm. Therefore, the area A of the entire hexagon is A = 6( 5 3) = cm. In general terms, if x represents a side of the regular polygon, and we form n triangles by drawing all radii of the regular polygon, then bh xa The area of one triangle = =. So for an n-sided regular polygon, the area A of the entire polygon (formed by n triangles) is xa nxa A= n = Since the perimeter P of the regular polygon can be represented by P = nx, we substitute this into the above equation and obtain nxa Pa A= = = 1 ap Notice that in both methods used above, we obtained the same general formula for the area of a regular polygon. This formula can be used as a shortcut instead of first dividing the regular polygon into triangles in order to find its area. The Area of a Regular Polygon If the apothem and perimeter of a regular polygon are denoted as a and P, respectively, then the area of the regular polygon is given by the formula: A= 1 ap

30 Examples Answer the following. 1. Use the formula for the area of a regular polygon to find the area of a regular hexagon with side length 16 in.. Find the area of an equilateral triangle with side length 10 in by using the following two methods: a) Use the formula for the area of a regular polygon. b) Find the height of the triangle and use the general formula for the area of any triangle. 3. Find the area of a square with side length 7 in by using the following two methods: a) Use the formula for the area of a regular polygon. b) Use the formula for the area of a square. Solutions: 1. The perimeter, P, of the regular hexagon is P = 6(16) = 96 in. The measure of one of its central angles is = 60. If an apothem is drawn, we can see that it forms a 30 o -60 o -90 o triangle, as shown in the enlarged view of the triangle below. Enlarged View 60 o a 30 o 16 in 16 in 60 o The shorter leg of the 30 o -60 o -90 o triangle has length 8 cm, since the apothem intersects the side at its midpoint. The length of the longer leg, a, is 3 times the length of the shorter leg, so a = 8 3 cm. We now have the information that we need to find the area of the regular hexagon: 1 1 ( )( ) A= ap= = in.

31 . a) The perimeter, P, of the equilateral triangle is P = 3(10) = 30 in. The measure of one of its central angles is =10. If an apothem is drawn, we can see that it forms a 30 o -60 o -90 o triangle, as shown in the enlarged view of the triangle below. Enlarged View 10 o a 60 o 30 o 10 in 10 in 5 in Focusing now on the 30 o -60 o -90 o triangle, the length of the shorter leg is a. Since the length of the longer leg is 3 times the length of the shorter leg, the length of the longer leg is a 3. We know that the length of the longer leg is 5 in (since the apothem intersects the side at its midpoint), so a 3 = 5. Solving for a and rationalizing the denominator, we obtain a = = = We now have the information that we need to find the area of the equilateral triangle: ( )( ) A= ap= 30 = = 5 3 in b) We will now find the height of the equilateral triangle and then use the 1 general formula A= bh for the area of a triangle. An equilateral triangle has three angles of equal measure. Since the sum of the measures of a triangle is 180 o, each angle of an equilateral triangle measures 60 o. We then draw an altitude of the triangle (i.e. its height), which forms a 30 o -60 o -90 o triangle, as shown below.

32 60 o h 30 o 60 o 60 o 60 o 5 in 10 in The shorter leg of the 30 o -60 o -90 o triangle has length 5 in, since the altitude intersects the base ( b = 10 in) at its midpoint. The length of the longer leg, h, is 3 times the length of the shorter leg, so h = 5 3 in. We can now find the area of the equilateral triangle by using the formula for the area of a triangle: 1 1 ( )( ) A= bh= = 5 3 in 3. a) The perimeter, P, of the square is P = 4(7) = 8 in. The measure of one of its central angles is 360 = 90. If an apothem is drawn, we can see that it 4 forms a 45 o -45 o -90 o triangle, as shown in the enlarged view of the triangle below. Enlarged View 90 o a 45 o 7 in 7 in 45 o 3.5 in Focusing now on the 45 o -45 o -90 o triangle, the legs of the triangle are congruent, so a = 3.5 in. We now have the information that we need to find the area of the square: 1 1 ( )( ) A= ap = = 49 in.

33 b) Since the side, s, of the square is 7 in, we can use the formula A= s to find the area of the square. A = s = 7 = 49 in In Example 3 above, it was significantly shorter to use the direct formula for the area of a square (part b) than to use the formula for a regular polygon (part a). In Example, the two methods were more equivalent. In Example 1, however, (the regular hexagon) there was only method available to find the area since we have not learned a formula for the area of a regular hexagon apart from the general formula for any regular polygon. Notice that the only detailed examples discussed in this section were the equilateral triangle, the square, and the regular hexagon. The reason for this is because those are the only regular polygons for which we can use special right triangles (30 o -60 o -90 o and 45 o - 45 o -90 o triangles) to find the length of the apothem. If we wanted to find the area, for example, or a regular pentagon (or any other regular polygon not mentioned above), we 1 could still use the formula A= ap to compute its area, but we would need to use Trigonometry to find the length of the apothem. (Refer to the upcoming unit on applications of Right Triangle Trigonometry for more details on how to do this.) Exercises Answer the following. Be sure to include units in each answer. (Figures may not be drawn to scale.) 1. Find the perimeter of a square with side length 100 cm.. Find the side length of a square with perimeter 100 cm. 3. Find the area of a square with side length 100 cm. 4. Find the side length of a square with area 100 cm. 5. Find the perimeter of an equilateral triangle with side length 6 in. 6. Find the side length of an equilateral triangle with perimeter 6 in. 7. Find the area of an equilateral triangle with side length 6 in. 8. Find the perimeter of a rectangle with length 7 m and width 5 m. 9. Find the area of a rectangle with length 7 m and width 5 m. 10. Find the width of a rectangle with length 8 in and perimeter 40 in. 11. Find the width of a rectangle with length 8 in and area 40 in. 1. Find the perimeter of a regular hexagon with side length 6 cm. 13. Find the area of a regular hexagon with side length 6 cm.

34 14. Find the side length of a regular hexagon with perimeter 6 cm. 15. If a photo frame has perimeter 54 in and width 1 in, find its length. 16. If a photo frame has area 54 in and length 1 in, find its width. 17. If a square has side length ( x + 3) ft, write an algebraic expression for the area of the square in terms of x. 18. If a rectangle has width ( x + 3) ft and length ( 7) x + ft, write an algebraic expression for the area of the rectangle in terms of x. 19. If a rectangle has width ( x + 3) ft, length ( 7) a) Find the value of x. b) Find the width. c) Find the length. 0. If a rectangle has width ( x + 3) ft, length ( 7) a) Find the value of x. b) Find the width. c) Find the length. 5 3 ft 1. If a rectangle has area ( x x ) in terms of x. x + ft, and area 165 ft, + + and width ( x + ) x + ft, and perimeter 104 ft, 1 ft, find the length. Find the perimeter of the following swimming pool. (Assume that all intersecting sides in the diagram are perpendicular to each other.) 15 m 0 m Pool 3 m 9 m 3. Find the area of the swimming pool in the diagram above. 4. Find the circumference of a circle with radius 16m. 5. Find the circumference of a circle with diameter 16m. 6. Find the area of a circle with radius 16m. 7. Find the area of a circle with diameter 16m.

35 8. Find the radius of a circle with area 16π m. 9. Find the radius of a circle with circumference 16π m. 30. Find the area of a circle with circumference 36π cm. 31. Find the circumference of a circle with area 36π cm. 3. Find the area of a parallelogram with base 5 cm and height 6 cm. 33. Find the area of a triangle with base 5 cm and height 6 cm. 34. Find the area of a trapezoid with bases 5 cm and 1 cm, and height 6 cm. 35. If a parallelogram has area 8 in and base 8 in, find the height. 36. If a triangle has area 8 in and base 8 in, find the height. 37. If a trapezoid has area 8 in and bases 8 in and 1 in, find the height. 38. Find the area of the following figures: a) 1 cm 7 cm b) 6 cm 11 cm 60 o c) 5 ft 9 ft 45 o

36 10 ft d) 8 ft 45 o e) 4 cm 7 cm 0 cm f) 8 m 60 o 13 m 8 m g) 7 m 45 o

### Area. Area Overview. Define: Area:

Define: Area: Area Overview Kite: Parallelogram: Rectangle: Rhombus: Square: Trapezoid: Postulates/Theorems: Every closed region has an area. If closed figures are congruent, then their areas are equal.

### Postulate 17 The area of a square is the square of the length of a. Postulate 18 If two figures are congruent, then they have the same.

Chapter 11: Areas of Plane Figures (page 422) 11-1: Areas of Rectangles (page 423) Rectangle Rectangular Region Area is measured in units. Postulate 17 The area of a square is the square of the length

### SURFACE AREA AND VOLUME

SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has

### Applications for Triangles

Not drawn to scale Applications for Triangles 1. 36 in. 40 in. 33 in. 1188 in. 2 69 in. 2 138 in. 2 1440 in. 2 2. 188 in. 2 278 in. 2 322 in. 2 none of these Find the area of a parallelogram with the given

### Chapter 7 Quiz. (1.) Which type of unit can be used to measure the area of a region centimeter, square centimeter, or cubic centimeter?

Chapter Quiz Section.1 Area and Initial Postulates (1.) Which type of unit can be used to measure the area of a region centimeter, square centimeter, or cubic centimeter? (.) TRUE or FALSE: If two plane

### Geometry Unit 6 Areas and Perimeters

Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose

### SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.

### Area of a triangle: The area of a triangle can be found with the following formula: You can see why this works with the following diagrams:

Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 You can see why this works with the following diagrams: h h b b Solve: Find the area of

### Student Outcomes. Lesson Notes. Classwork. Exercises 1 3 (4 minutes)

Student Outcomes Students give an informal derivation of the relationship between the circumference and area of a circle. Students know the formula for the area of a circle and use it to solve problems.

### 2006 Geometry Form A Page 1

2006 Geometry Form Page 1 1. he hypotenuse of a right triangle is 12" long, and one of the acute angles measures 30 degrees. he length of the shorter leg must be: () 4 3 inches () 6 3 inches () 5 inches

### Conjectures. Chapter 2. Chapter 3

Conjectures Chapter 2 C-1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C-2 Vertical Angles Conjecture If two angles are vertical

### Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.

Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 9-1.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles

### Conjectures for Geometry for Math 70 By I. L. Tse

Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:

### 43 Perimeter and Area

43 Perimeter and Area Perimeters of figures are encountered in real life situations. For example, one might want to know what length of fence will enclose a rectangular field. In this section we will study

### 12 Surface Area and Volume

12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids

### Area of Parallelograms, Triangles, and Trapezoids (pages 314 318)

Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base

### GAP CLOSING. 2D Measurement. Intermediate / Senior Student Book

GAP CLOSING 2D Measurement Intermediate / Senior Student Book 2-D Measurement Diagnostic...3 Areas of Parallelograms, Triangles, and Trapezoids...6 Areas of Composite Shapes...14 Circumferences and Areas

### Geometry Notes PERIMETER AND AREA

Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

### Chapter 11. Areas of Plane Figures You MUST draw diagrams and show formulas for every applicable homework problem!

Chapter 11 Areas of Plane Figures You MUST draw diagrams and show formulas for every applicable homework problem! Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret

### 56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.

6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which

### Geometry Final Exam Review Worksheet

Geometry Final xam Review Worksheet (1) Find the area of an equilateral triangle if each side is 8. (2) Given the figure to the right, is tangent at, sides as marked, find the values of x, y, and z please.

### 9 Area, Perimeter and Volume

9 Area, Perimeter and Volume 9.1 2-D Shapes The following table gives the names of some 2-D shapes. In this section we will consider the properties of some of these shapes. Rectangle All angles are right

### of surface, 569-571, 576-577, 578-581 of triangle, 548 Associative Property of addition, 12, 331 of multiplication, 18, 433

Absolute Value and arithmetic, 730-733 defined, 730 Acute angle, 477 Acute triangle, 497 Addend, 12 Addition associative property of, (see Commutative Property) carrying in, 11, 92 commutative property

### Characteristics of the Four Main Geometrical Figures

Math 40 9.7 & 9.8: The Big Four Square, Rectangle, Triangle, Circle Pre Algebra We will be focusing our attention on the formulas for the area and perimeter of a square, rectangle, triangle, and a circle.

### Parallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.

CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes

### Area of a triangle: The area of a triangle can be found with the following formula: 1. 2. 3. 12in

Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 Solve: Find the area of each triangle. 1. 2. 3. 5in4in 11in 12in 9in 21in 14in 19in 13in

### Grade 3 Core Standard III Assessment

Grade 3 Core Standard III Assessment Geometry and Measurement Name: Date: 3.3.1 Identify right angles in two-dimensional shapes and determine if angles are greater than or less than a right angle (obtuse

### GEOMETRY CONCEPT MAP. Suggested Sequence:

CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons

### Perimeter is the length of the boundary of a two dimensional figure.

Section 2.2: Perimeter and Area Perimeter is the length of the boundary of a two dimensional figure. The perimeter of a circle is called the circumference. The perimeter of any two dimensional figure whose

### Algebra Geometry Glossary. 90 angle

lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

### CSU Fresno Problem Solving Session. Geometry, 17 March 2012

CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news

### Calculating Area, Perimeter and Volume

Calculating Area, Perimeter and Volume You will be given a formula table to complete your math assessment; however, we strongly recommend that you memorize the following formulae which will be used regularly

### Area of Parallelograms (pages 546 549)

A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

### GAP CLOSING. Volume and Surface Area. Intermediate / Senior Student Book

GAP CLOSING Volume and Surface Area Intermediate / Senior Student Book Volume and Surface Area Diagnostic...3 Volumes of Prisms...6 Volumes of Cylinders...13 Surface Areas of Prisms and Cylinders...18

### Show that when a circle is inscribed inside a square the diameter of the circle is the same length as the side of the square.

Week & Day Week 6 Day 1 Concept/Skill Perimeter of a square when given the radius of an inscribed circle Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional

### Geometry Regents Review

Name: Class: Date: Geometry Regents Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. If MNP VWX and PM is the shortest side of MNP, what is the shortest

### Area is a measure of how much space is occupied by a figure. 1cm 1cm

Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number

### GAP CLOSING. 2D Measurement GAP CLOSING. Intermeditate / Senior Facilitator s Guide. 2D Measurement

GAP CLOSING 2D Measurement GAP CLOSING 2D Measurement Intermeditate / Senior Facilitator s Guide 2-D Measurement Diagnostic...4 Administer the diagnostic...4 Using diagnostic results to personalize interventions...4

### 2nd Semester Geometry Final Exam Review

Class: Date: 2nd Semester Geometry Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of an amusement park created a circular

### EVERY DAY COUNTS CALENDAR MATH 2005 correlated to

EVERY DAY COUNTS CALENDAR MATH 2005 correlated to Illinois Mathematics Assessment Framework Grades 3-5 E D U C A T I O N G R O U P A Houghton Mifflin Company YOUR ILLINOIS GREAT SOURCE REPRESENTATIVES:

### 12 Surface Area and Volume

CHAPTER 12 Surface Area and Volume Chapter Outline 12.1 EXPLORING SOLIDS 12.2 SURFACE AREA OF PRISMS AND CYLINDERS 12.3 SURFACE AREA OF PYRAMIDS AND CONES 12.4 VOLUME OF PRISMS AND CYLINDERS 12.5 VOLUME

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of

### CK-12 Geometry: Parts of Circles and Tangent Lines

CK-12 Geometry: Parts of Circles and Tangent Lines Learning Objectives Define circle, center, radius, diameter, chord, tangent, and secant of a circle. Explore the properties of tangent lines and circles.

### Shape Dictionary YR to Y6

Shape Dictionary YR to Y6 Guidance Notes The terms in this dictionary are taken from the booklet Mathematical Vocabulary produced by the National Numeracy Strategy. Children need to understand and use

### Quick Reference ebook

This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference ebook Click on Contents or Index in the left panel to locate a topic. The math facts listed

### Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees

Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Apex in a pyramid or cone, the vertex opposite the base; in

### CCGPS UNIT 3 Semester 1 ANALYTIC GEOMETRY Page 1 of 32. Circles and Volumes Name:

GPS UNIT 3 Semester 1 NLYTI GEOMETRY Page 1 of 3 ircles and Volumes Name: ate: Understand and apply theorems about circles M9-1.G..1 Prove that all circles are similar. M9-1.G.. Identify and describe relationships

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### 12-1 Representations of Three-Dimensional Figures

Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

### Grade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013

Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is

### Chapters 6 and 7 Notes: Circles, Locus and Concurrence

Chapters 6 and 7 Notes: Circles, Locus and Concurrence IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of

### Geometry and Measurement

The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

### Illinois State Standards Alignments Grades Three through Eleven

Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other

### Chapter 8 Geometry We will discuss following concepts in this chapter.

Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### Geometry Enduring Understandings Students will understand 1. that all circles are similar.

High School - Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,

### B = 1 14 12 = 84 in2. Since h = 20 in then the total volume is. V = 84 20 = 1680 in 3

45 Volume Surface area measures the area of the two-dimensional boundary of a threedimensional figure; it is the area of the outside surface of a solid. Volume, on the other hand, is a measure of the space

### Tangent Properties. Line m is a tangent to circle O. Point T is the point of tangency.

CONDENSED LESSON 6.1 Tangent Properties In this lesson you will Review terms associated with circles Discover how a tangent to a circle and the radius to the point of tangency are related Make a conjecture

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS WITHOUT A CALCULATOR!

DETAILED SOLUTIONS AND CONCEPTS - SIMPLE GEOMETRIC FIGURES Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! YOU MUST

### Circumference Pi Regular polygon. Dates, assignments, and quizzes subject to change without advance notice.

Name: Period GPreAP UNIT 14: PERIMETER AND AREA I can define, identify and illustrate the following terms: Perimeter Area Base Height Diameter Radius Circumference Pi Regular polygon Apothem Composite

### Angles that are between parallel lines, but on opposite sides of a transversal.

GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

### Unit 3 Practice Test. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: lass: ate: I: Unit 3 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. The radius, diameter, or circumference of a circle is given. Find

### Geometry Course Summary Department: Math. Semester 1

Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### MATH STUDENT BOOK. 6th Grade Unit 8

MATH STUDENT BOOK 6th Grade Unit 8 Unit 8 Geometry and Measurement MATH 608 Geometry and Measurement INTRODUCTION 3 1. PLANE FIGURES 5 PERIMETER 5 AREA OF PARALLELOGRAMS 11 AREA OF TRIANGLES 17 AREA OF

### Solids. Objective A: Volume of a Solids

Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2015 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2015 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications

### 7 th Grade Study guide IV Partial Remember to practice the constructions that are not part of this guide.

7 th Grade Study guide IV Partial Remember to practice the constructions that are not part of this guide. 1. Which figure shows one point? a. S R c. D C b. Q d. F G 2. Which name describes the line? G

### Circumference of a Circle

Circumference of a Circle A circle is a shape with all points the same distance from the center. It is named by the center. The circle to the left is called circle A since the center is at point A. If

### 11.3 Curves, Polygons and Symmetry

11.3 Curves, Polygons and Symmetry Polygons Simple Definition A shape is simple if it doesn t cross itself, except maybe at the endpoints. Closed Definition A shape is closed if the endpoints meet. Polygon

### Conjunction is true when both parts of the statement are true. (p is true, q is true. p^q is true)

Mathematical Sentence - a sentence that states a fact or complete idea Open sentence contains a variable Closed sentence can be judged either true or false Truth value true/false Negation not (~) * Statement

### 4. How many integers between 2004 and 4002 are perfect squares?

5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

### Integrated Math Concepts Module 10. Properties of Polygons. Second Edition. Integrated Math Concepts. Solve Problems. Organize. Analyze. Model.

Solve Problems Analyze Organize Reason Integrated Math Concepts Model Measure Compute Communicate Integrated Math Concepts Module 1 Properties of Polygons Second Edition National PASS Center 26 National

### Calculating Perimeter

Calculating Perimeter and Area Formulas are equations used to make specific calculations. Common formulas (equations) include: P = 2l + 2w perimeter of a rectangle A = l + w area of a square or rectangle

### Definitions, Postulates and Theorems

Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven

### GEOMETRY COMMON CORE STANDARDS

1st Nine Weeks Experiment with transformations in the plane G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point,

### 1-6 Two-Dimensional Figures. Name each polygon by its number of sides. Then classify it as convex or concave and regular or irregular.

Stop signs are constructed in the shape of a polygon with 8 sides of equal length The polygon has 8 sides A polygon with 8 sides is an octagon All sides of the polygon are congruent and all angles are

### MENSURATION. Definition

MENSURATION Definition 1. Mensuration : It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids. 2. Plane Mensuration : It deals with the sides, perimeters

### Target To know the properties of a rectangle

Target To know the properties of a rectangle (1) A rectangle is a 3-D shape. (2) A rectangle is the same as an oblong. (3) A rectangle is a quadrilateral. (4) Rectangles have four equal sides. (5) Rectangles

### Geometry of 2D Shapes

Name: Geometry of 2D Shapes Answer these questions in your class workbook: 1. Give the definitions of each of the following shapes and draw an example of each one: a) equilateral triangle b) isosceles

### FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST. Mathematics Reference Sheets. Copyright Statement for this Assessment and Evaluation Services Publication

FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST Mathematics Reference Sheets Copyright Statement for this Assessment and Evaluation Services Publication Authorization for reproduction of this document is hereby

### Circle Name: Radius: Diameter: Chord: Secant:

12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane

### Surface Area and Volume Cylinders, Cones, and Spheres

Surface Area and Volume Cylinders, Cones, and Spheres Michael Fauteux Rosamaria Zapata CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable

### Georgia Online Formative Assessment Resource (GOFAR) AG geometry domain

AG geometry domain Name: Date: Copyright 2014 by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 36 ) 1. Amy drew a circle graph to represent

### MEASUREMENTS. U.S. CUSTOMARY SYSTEM OF MEASUREMENT LENGTH The standard U.S. Customary System units of length are inch, foot, yard, and mile.

MEASUREMENTS A measurement includes a number and a unit. 3 feet 7 minutes 12 gallons Standard units of measurement have been established to simplify trade and commerce. TIME Equivalences between units

### Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms.

Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game

### Geometry. Higher Mathematics Courses 69. Geometry

The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and

### Session 7 Circles and Pi (π)

Key Terms in This Session Session 7 Circles and Pi (π) Previously Introduced accuracy area precision scale factor similar figures New in This Session circumference diameter irrational number perimeter

### Lesson 21. Circles. Objectives

Student Name: Date: Contact Person Name: Phone Number: Lesson 1 Circles Objectives Understand the concepts of radius and diameter Determine the circumference of a circle, given the diameter or radius Determine

### Tallahassee Community College PERIMETER

Tallahassee Community College 47 PERIMETER The perimeter of a plane figure is the distance around it. Perimeter is measured in linear units because we are finding the total of the lengths of the sides

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Thursday, August 13, 2009 8:30 to 11:30 a.m., only.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Thursday, August 13, 2009 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of your

### Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Name: Class: Date: ID: A Q3 Geometry Review Multiple Choice Identify the choice that best completes the statement or answers the question. Graph the image of each figure under a translation by the given

### Chapter 4: Area, Perimeter, and Volume. Geometry Assessments

Chapter 4: Area, Perimeter, and Volume Geometry Assessments Area, Perimeter, and Volume Introduction The performance tasks in this chapter focus on applying the properties of triangles and polygons to

### CAMI Education linked to CAPS: Mathematics

- 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to

### SGS4.3 Stage 4 Space & Geometry Part A Activity 2-4

SGS4.3 Stage 4 Space & Geometry Part A Activity 2-4 Exploring triangles Resources required: Each pair students will need: 1 container (eg. a rectangular plastic takeaway container) 5 long pipe cleaners

### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m.

GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 28, 2015 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any

### ACTIVITY: Finding a Formula Experimentally. Work with a partner. Use a paper cup that is shaped like a cone.

8. Volumes of Cones How can you find the volume of a cone? You already know how the volume of a pyramid relates to the volume of a prism. In this activity, you will discover how the volume of a cone relates