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

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

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

### 10.1: Areas of Parallelograms and Triangles

10.1: Areas of Parallelograms and Triangles Important Vocabulary: By the end of this lesson, you should be able to define these terms: Base of a Parallelogram, Altitude of a Parallelogram, Height of a

### 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

### Perimeter and Area. An artist uses perimeter and area to determine the amount of materials it takes to produce a piece such as this.

UNIT 10 Perimeter and Area An artist uses perimeter and area to determine the amount of materials it takes to produce a piece such as this. 3 UNIT 10 PERIMETER AND AREA You can find geometric shapes in

### Perimeter and area formulas for common geometric figures:

Lesson 10.1 10.: Perimeter and Area of Common Geometric Figures Focused Learning Target: I will be able to Solve problems involving perimeter and area of common geometric figures. Compute areas of rectangles,

### Geometry. Geometry is the study of shapes and sizes. The next few pages will review some basic geometry facts. Enjoy the short lesson on geometry.

Geometry Introduction: We live in a world of shapes and figures. Objects around us have length, width and height. They also occupy space. On the job, many times people make decision about what they know

### 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

### 11-4 Areas of Regular Polygons and Composite Figures

1. In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. Center: point F, radius:, apothem:,

### LEVEL G, SKILL 1. Answers Be sure to show all work.. Leave answers in terms of ϖ where applicable.

Name LEVEL G, SKILL 1 Class Be sure to show all work.. Leave answers in terms of ϖ where applicable. 1. What is the area of a triangle with a base of 4 cm and a height of 6 cm? 2. What is the sum of the

### 1. A person has 78 feet of fencing to make a rectangular garden. What dimensions will use all the fencing with the greatest area?

1. A person has 78 feet of fencing to make a rectangular garden. What dimensions will use all the fencing with the greatest area? (a) 20 ft x 19 ft (b) 21 ft x 18 ft (c) 22 ft x 17 ft 2. Which conditional

### Study Guide. 6.g.1 Find the area of triangles, quadrilaterals, and other polygons. Note: Figure is not drawn to scale.

Study Guide Name Test date 6.g.1 Find the area of triangles, quadrilaterals, and other polygons. 1. Note: Figure is not drawn to scale. If x = 14 units and h = 6 units, then what is the area of the triangle

### 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

### Geometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources:

Geometry Honors: Extending 2 Dimensions into 3 Dimensions Semester 2, Unit 5: Activity 30 Resources: SpringBoard- Geometry Online Resources: Geometry Springboard Text Unit Overview In this unit students

### 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (ft) 1 mile (mi) = 5280 feet (ft) Replace 1 with 1 ft/12 in. 1ft

2 MODULE 6. GEOMETRY AND UNIT CONVERSION 6a Applications The most common units of length in the American system are inch, foot, yard, and mile. Converting from one unit of length to another is a requisite

### Geometry Unit 7 (Textbook Chapter 9) Solving a right triangle: Find all missing sides and all missing angles

Geometry Unit 7 (Textbook Chapter 9) Name Objective 1: Right Triangles and Pythagorean Theorem In many geometry problems, it is necessary to find a missing side or a missing angle of a right triangle.

### 10.1 Areas of Quadrilaterals and triangles

10.1 Areas of Quadrilaterals and triangles BASE AND HEIGHT MUST FORM A RIGHT ANGLE!! Draw the diagram, write the formula and SHOW YOUR WORK! FIND THE AREA OF THE FOLLOWING:. A rectangle with one side 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.

### Integrated Algebra: Geometry

Integrated Algebra: Geometry Topics of Study: o Perimeter and Circumference o Area Shaded Area Composite Area o Volume o Surface Area o Relative Error Links to Useful Websites & Videos: o Perimeter and

### (a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units

1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units

### 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

### PERIMETERS AND AREAS

PERIMETERS AND AREAS 1. PERIMETER OF POLYGONS The Perimeter of a polygon is the distance around the outside of the polygon. It is the sum of the lengths of all the sides. Examples: The perimeter of this

### 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

### 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

### Topics Covered on Geometry Placement Exam

Topics Covered on Geometry Placement Exam - Use segments and congruence - Use midpoint and distance formulas - Measure and classify angles - Describe angle pair relationships - Use parallel lines and transversals

### Unit 3: Triangle Bisectors and Quadrilaterals

Unit 3: Triangle Bisectors and Quadrilaterals Unit Objectives Identify triangle bisectors Compare measurements of a triangle Utilize the triangle inequality theorem Classify Polygons Apply the properties

### Name: Class: Date: Geometry Chapter 3 Review

Name: Class: Date: ID: A Geometry Chapter 3 Review. 1. The area of a rectangular field is 6800 square meters. If the width of the field is 80 meters, what is the perimeter of the field? Draw a diagram

### 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:

### 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

### Signs, Signs, Every Place There Are Signs! Area of Regular Polygons p. 171 Boundary Lines Area of Parallelograms and Triangles p.

C H A P T E R Perimeter and Area Regatta is another word for boat race. In sailing regattas, sailboats compete on courses defined by marks or buoys. These courses often start and end at the same mark,

### Perfume Packaging. Ch 5 1. Chapter 5: Solids and Nets. Chapter 5: Solids and Nets 279. The Charles A. Dana Center. Geometry Assessments Through

Perfume Packaging Gina would like to package her newest fragrance, Persuasive, in an eyecatching yet cost-efficient box. The Persuasive perfume bottle is in the shape of a regular hexagonal prism 10 centimeters

### 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

### Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade

Content Area: GEOMETRY Grade 9 th Quarter 1 st Curso Serie Unidade Standards/Content Padrões / Conteúdo Learning Objectives Objetivos de Aprendizado Vocabulary Vocabulário Assessments Avaliações Resources

### Areas of Rectangles and Parallelograms

CONDENSED LESSON 8.1 Areas of Rectangles and Parallelograms In this lesson you will Review the formula for the area of a rectangle Use the area formula for rectangles to find areas of other shapes Discover

### CONJECTURES - Discovering Geometry. Chapter 2

CONJECTURES - Discovering Geometry Chapter C-1 Linear Pair Conjecture - If two angles form a linear pair, then the measures of the angles add up to 180. C- Vertical Angles Conjecture - If two angles are

### 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

### A. Areas of Parallelograms 1. If a parallelogram has an area of A square units, a base of b units, and a height of h units, then A = bh.

Geometry - Areas of Parallelograms A. Areas of Parallelograms. If a parallelogram has an area of A square units, a base of b units, and a height of h units, then A = bh. A B Ex: See how VDFA V CGB so rectangle

### Review for Final - Geometry B

Review for Final - Geometry B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A model is made of a car. The car is 4 meters long and the model is 7 centimeters

### 12-4 Volumes of Prisms and Cylinders. Find the volume of each prism. The volume V of a prism is V = Bh, where B is the area of a base and h

Find the volume of each prism. The volume V of a prism is V = Bh, where B is the area of a base and h The volume is 108 cm 3. The volume V of a prism is V = Bh, where B is the area of a base and h the

### BASIC GEOMETRY GLOSSARY

BASIC GEOMETRY GLOSSARY Acute angle An angle that measures between 0 and 90. Examples: Acute triangle A triangle in which each angle is an acute angle. Adjacent angles Two angles next to each other that

### 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

# 30-60 right triangle, 441-442, 684 A Absolute value, 59 Acute angle, 77, 669 Acute triangle, 178 Addition Property of Equality, 86 Addition Property of Inequality, 258 Adjacent angle, 109, 669 Adjacent

### Centroid: The point of intersection of the three medians of a triangle. Centroid

Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

### 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

### The Area is the width times the height: Area = w h

Geometry Handout Rectangle and Square Area of a Rectangle and Square (square has all sides equal) The Area is the width times the height: Area = w h Example: A rectangle is 6 m wide and 3 m high; what

### 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

### 12-4 Volumes of Prisms and Cylinders. Find the volume of each prism.

Find the volume of each prism. 3. the oblique rectangular prism shown at the right 1. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. If two solids have

### GEOMETRY FINAL EXAM REVIEW

GEOMETRY FINL EXM REVIEW I. MTHING reflexive. a(b + c) = ab + ac transitive. If a = b & b = c, then a = c. symmetric. If lies between and, then + =. substitution. If a = b, then b = a. distributive E.

### 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

### 3. If AC = 12, CD = 9 and BE = 3, find the area of trapezoid BCDE. (Mathcounts Handbooks)

EXERCISES: Triangles 1 1. The perimeter of an equilateral triangle is units. How many units are in the length 27 of one side? (Mathcounts Competitions) 2. In the figure shown, AC = 4, CE = 5, DE = 3, and

### 10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres. 10.4 Day 1 Warm-up

10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres 10.4 Day 1 Warm-up 1. Which identifies the figure? A rectangular pyramid B rectangular prism C cube D square pyramid 3. A polyhedron

### MATH 139 FINAL EXAM REVIEW PROBLEMS

MTH 139 FINL EXM REVIEW PROLEMS ring a protractor, compass and ruler. Note: This is NOT a practice exam. It is a collection of problems to help you review some of the material for the exam and to practice

### 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

### Su.a Supported: Identify Determine if polygons. polygons with all sides have all sides and. and angles equal angles equal (regular)

MA.912.G.2 Geometry: Standard 2: Polygons - Students identify and describe polygons (triangles, quadrilaterals, pentagons, hexagons, etc.), using terms such as regular, convex, and concave. They find measures

### 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

Unit 6: POLYGONS Are these polygons? Justify your answer by explaining WHY or WHY NOT??? a) b) c) Yes or No Why/Why not? Yes or No Why/Why not? Yes or No Why/Why not? a) What is a CONCAVE polygon? Use

### 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

### 8-3 Perimeter and Circumference

Learn to find the perimeter of a polygon and the circumference of a circle. 8-3 Perimeter Insert Lesson and Title Circumference Here perimeter circumference Vocabulary The distance around a geometric figure

### Math 6: Unit 7: Geometry Notes 2-Dimensional Figures

Math 6: Unit 7: Geometry Notes -Dimensional Figures Prep for 6.G.A.1 Classifying Polygons A polygon is defined as a closed geometric figure formed by connecting line segments endpoint to endpoint. Polygons

### Geometry Chapter 9 Extending Perimeter, Circumference, and Area

Geometry Chapter 9 Extending Perimeter, Circumference, and Area Lesson 1 Developing Formulas for Triangles and Quadrilaterals Learning Target (LT-1) Solve problems involving the perimeter and area of triangles

### 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 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

Unit 8 Quadrilaterals Academic Geometry Spring 2014 Name Teacher Period 1 2 3 Unit 8 at a glance Quadrilaterals This unit focuses on revisiting prior knowledge of polygons and extends to formulate, test,

### Surface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry

Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel

### 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

### Grade 3 Math Expressions Vocabulary Words

Grade 3 Math Expressions Vocabulary Words Unit 1, Book 1 Place Value and Multi-Digit Addition and Subtraction OSPI words not used in this unit: add, addition, number, more than, subtract, subtraction,

### II. Geometry and Measurement

II. Geometry and Measurement The Praxis II Middle School Content Examination emphasizes your ability to apply mathematical procedures and algorithms to solve a variety of problems that span multiple mathematics

### 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

### 11-2 Areas of Trapezoids, Rhombi, and Kites. Find the area of each trapezoid, rhombus, or kite. 1. SOLUTION: 2. SOLUTION: 3.

Find the area of each trapezoid, rhombus, or kite. 1. 2. 3. esolutions Manual - Powered by Cognero Page 1 4. OPEN ENDED Suki is doing fashion design at 4-H Club. Her first project is to make a simple A-line

### A factor is a whole number that. Name 6 different quadrilaterals. The radius of a circle. What is an axis or a line of symmetry in a 2-D shape?

BOND HOW TO DO 11+ MATHS MATHS FACTS CARDS 1 2 3 4 A factor is a whole number that Name 6 different quadrilaterals. The radius of a circle is What is an axis or a line of symmetry in a 2-D shape? 5 6 7

### 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

### Geometry. Unit 6. Quadrilaterals. Unit 6

Geometry Quadrilaterals Properties of Polygons Formed by three or more consecutive segments. The segments form the sides of the polygon. Each side intersects two other sides at its endpoints. The intersections

### Special case: Square. The same formula works, but you can also use A= Side x Side or A= (Side) 2

Geometry Chapter 11/12 Review Shape: Rectangle Formula A= Base x Height Special case: Square. The same formula works, but you can also use A= Side x Side or A= (Side) 2 Height = 6 Base = 8 Area = 8 x 6

### 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

### Geometry Chapter 9 Extending Perimeter, Circumference, and Area

Geometry Chapter 9 Extending Perimeter, Circumference, and Area Lesson 1 Developing Formulas for Triangles and Quadrilaterals Learning Targets LT9-1: Solve problems involving the perimeter and area of

### Surface Area and Volume Nets to Prisms

Surface Area and Volume Nets to Prisms 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 version

### Geometry, Final Review Packet

Name: Geometry, Final Review Packet I. Vocabulary match each word on the left to its definition on the right. Word Letter Definition Acute angle A. Meeting at a point Angle bisector B. An angle with a

### Geo - CH9 Practice Test

Geo - H9 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Find the area of the parallelogram. a. 35 in 2 c. 21 in 2 b. 14 in 2 d. 28 in 2 2.

### Geometry Chapter 12. Volume. Surface Area. Similar shapes ratio area & volume

Geometry Chapter 12 Volume Surface Area Similar shapes ratio area & volume Date Due Section Topics Assignment Written Exercises 12.1 Prisms Altitude Lateral Faces/Edges Right vs. Oblique Cylinders 12.3

### 12-2 Surface Areas of Prisms and Cylinders. 1. Find the lateral area of the prism. SOLUTION: ANSWER: in 2

1. Find the lateral area of the prism. 3. The base of the prism is a right triangle with the legs 8 ft and 6 ft long. Use the Pythagorean Theorem to find the length of the hypotenuse of the base. 112.5

### Final Review Problems Geometry AC Name

Final Review Problems Geometry Name SI GEOMETRY N TRINGLES 1. The measure of the angles of a triangle are x, 2x+6 and 3x-6. Find the measure of the angles. State the theorem(s) that support your equation.

### 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:

### Sum of the interior angles of a n-sided Polygon = (n-2) 180

5.1 Interior angles of a polygon Sides 3 4 5 6 n Number of Triangles 1 Sum of interiorangles 180 Sum of the interior angles of a n-sided Polygon = (n-2) 180 What you need to know: How to use the formula

### 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

### 10-4 Inscribed Angles. Find each measure. 1.

Find each measure. 1. 3. 2. intercepted arc. 30 Here, is a semi-circle. So, intercepted arc. So, 66 4. SCIENCE The diagram shows how light bends in a raindrop to make the colors of the rainbow. If, what

### Archdiocese of Washington Catholic Schools Academic Standards Mathematics

5 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions,

### 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

### 10.1 Geometry Areas of Parallelograms, Triangles and Heron s Formula

Name Due Date 4/10 https://www.youtube.com/watch?v=eh5zawhrioo 10.1 Geometry Areas of Parallelograms, Triangles and Heron s Formula Area of a Rectangle: A= Area of a Square: A= Area of a Parallelogram:

### 1.7 Find Perimeter, Circumference,

.7 Find Perimeter, Circumference, and rea Goal p Find dimensions of polygons. Your Notes FORMULS FOR PERIMETER P, RE, ND CIRCUMFERENCE C Square Rectangle side length s length l and width w P 5 P 5 s 5

### 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

### Junior Math Circles November 18, D Geometry II

1 University of Waterloo Faculty of Mathematics Junior Math Circles November 18, 009 D Geometry II Centre for Education in Mathematics and Computing Two-dimensional shapes have a perimeter and an area.