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


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1 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
2 You can find geometric shapes in art. Whether determining the amount of leading or the amount of glass for a piece of stainedglass art, stainedglass artists need to understand perimeter and area to solve many practical problems. Big Ideas Several useful aspects of every geometric figure that can be measured, calculated, or approximated. A segment has a finite length that can be measured. Area is a measure of how much material is needed to cover a plane figure. Many problems can be solved by using the properties of angles, triangles, and circles. Unit Topics Types of Polygons Perimeter Areas of Rectangles and Triangles Special Quadrilaterals Areas of Special Quadrilaterals Circumference Areas of Circles PERIMETER AND AREA 35
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4 Types of Polygons Some geometric shapes are polygons. DEFINITIONS A polygon is a closed figure formed by three or more line segments in a plane, such that each line segment intersects two other line segments at their endpoints only. The endpoints are called vertices (the singular is vertex) and the segments are called sides. polygons not polygons Naming Polygons A polygon is named by its number of sides. An nsided polygon is called an ngon. Therefore, a polygon with 16 sides is a 16gon. For fewer numbers of sides, the following terms are commonly used. 3 sides: triangle 7 sides: heptagon sides: quadrilateral 8 sides: octagon 5 sides: pentagon 9 sides: nonagon 6 sides: hexagon 10 sides: decagon TIP To remember the names of polygons, think of words with the same prefixes, such as tricycle, octopus, and decade. Example 1 Name each polygon. A B Solution pentagon C Solution quadrilateral D Solution octagon Solution 0gon TYPES OF POLYGONS 37
5 Describing Polygons Tick marks indicate congruent sides and arcs indicate congruent angles. If a polygon is equiangular, all of its angles are congruent. If a polygon is equilateral, all of its sides are congruent. If a polygon is regular, it is both equiangular and equilateral. Example Determine whether each polygon is equiangular, equilateral, regular, or none of these. A B Solution equiangular C Solution equilateral D Solution regular E Solution none of these F Solution none of these Solution equilateral 38 UNIT 10 PERIMETER AND AREA
6 Example 3 The vertices of a polygon are given. Plot and connect the points in the order given. Determine if the polygon appears to be equiangular, equilateral, regular, or none of these. A (1, 1), (1, 3), (3, 3), (3, 1). Solution y (1, 3) (3, 3) (1, 1) (3, 1) x Plot and connect the points. All sides are congruent. All the angles are right angles. The quadrilateral is regular. B ( 3, ), ( 3, 1), (3, 1), (3, ). y ( 3, 1) (3, 1) ( 3, ) 3 (3, ) 5 5 x Plot and connect the points. Both pairs of opposite sides are congruent. All the angles are right angles. The quadrilateral is equiangular. Application: Sports Example The infield of a baseball diamond is bounded by the shape of a regular quadrilateral. At each vertex is a base. The total distance around the infield boundary is 360 feet. What is the distance between each base? Solution A figure that is regular is equilateral, so all the sides of the quadrilateral have the same length. Divide the total distance by the number of sides. 360 = 90 The distance between each pair of consecutive bases is 90 feet. THINK ABOUT IT A regular quadrilateral is also called a square. TYPES OF POLYGONS 39
7 Problem Set Name each polygon Determine whether each polygon is equiangular, equilateral, regular, or none of these. 9. cm 13. cm cm 10. cm cm cm 1. cm cm UNIT 10 PERIMETER AND AREA
8 The vertices of a polygon are given. Plot and connect the points in the order given. Determine if the polygon appears to be equiangular, equilateral, regular, or none of these. 17. (1, ), (, ), (, 5), (1, 5) 18. (1, ), (, 1), ( 5, ), (, 3) Write answers in complete sentences. 19. (6, 1), (, 3), (6, 6) 0. ( 7, 6), ( 5, 6), (, 3), ( 7, 3) 1. Why is a circle not a polygon?. Which polygon best represents a yield sign? 3. If a polygon has n sides, how many vertices does it have?. What is the measure of each angle of a regular octagon if the sum of the measures of all the angles is 1080? 5. The sum of the measures of the angles of a regular polygon is 30. One of the angles measures 16. Name the polygon. 6. What is the distance around a regular heptagon if each side is 11 inches long? 7. A clock is shaped like a regular 1gon. The sum of all the angle measures is What is the measure of each angle? * * 8. Challenge The sum of the measures of the angles of a quadrilateral is 360. Two of the angles are right angles. One of the remaining two angles is twice the measure of the other. What is the measure of the smallest angle in the quadrilateral? What is the measure of the largest angle? 9. Challenge Connecting one vertex to the other vertices in a polygon forms triangles. The example below shows how to form 3 nonoverlapping triangles inside a pentagon. How many nonoverlapping triangles are formed by connecting one vertex to the other nonadjacent vertices of a regular hexagon? a regular octagon? a regular ngon? TYPES OF POLYGONS 351
9 Perimeter The sides of a polygon form the boundary of the figure. DEFINITION Perimeter is the distance around a figure. The perimeter P of a polygon is the sum of the lengths of all its sides. When all sides of a polygon are congruent, as with a regular polygon, you can multiply the length of one side by the number of sides to find the polygon s perimeter. REMEMBER A regular polygon is equilateral (all sides congruent) and equiangular (all angles congruent). Finding the Perimeter of a Regular ngon A regular polygon has sides that are the same length, so you can use multiplication to find its perimeter. PERIMETER OF A REGULAR ngon The perimeter of a regular polygon with n sides each with length s is P = ns. Example 1 A Each side of a regular hexagon is 1 cm long. Find the perimeter of the hexagon. Solution A hexagon has 6 sides. P = ns Write the formula. = 6 1 Substitute 6 for n and 1 for s. = 7 Multiply. The perimeter is 7 cm. 35 UNIT 10 PERIMETER AND AREA
10 B Find the perimeter of an equilateral triangle if a side length is 3.5 feet. Solution A triangle has 3 sides. P = ns Write the formula. = Substitute 3 for n and 3.5 for s. = 10.5 Multiply. The perimeter is 10.5 feet. Finding the Perimeter of a Rectangle The opposite sides of a rectangle are congruent. PERIMETER OF A RECTANGLE The perimeter of a rectangle with width w and length l is w THINK ABOUT IT You can also write the formula as P = l + l + w + w or P = (l + w). P = l + w. l Example Find the perimeter of the rectangle. 6 mm 1 mm Solution Use the formula. P = l + w Write the formula. = Substitute 1 for l and 6 for w. = Multiply. = 0 Add. The perimeter is 0 millimeters. THINK ABOUT IT You would get the same perimeter by substituting 6 for l and 1 for w. Finding Missing Lengths You can use perimeter to find a missing side length. Example 3 A The perimeter of the rectangle is 68 km. The length of the rectangle is 3 km. What is the width of the rectangle? 3 km (continued) PERIMETER 353
11 Solution Substitute the known information into P = l + w. Then solve for w. P = l + w Write the formula. 68 = 3 + w Substitute 68 for P and 3 for l. 68 = 6 + w Multiply. = w Subtract 6 from both sides. 11 = w Divide both sides by. The width is 11 kilometers. Check P = l + w = = 6 + = 68. B The perimeter of a square is 36 meters. What is the length of each side of the square? Solution A square is a regular quadrilateral. It has congruent sides. Substitute the known information into P = s and solve for s. P = s Write the formula. 36 = s Substitute 36 for P. 9 = s Divide both sides by. Each side has a length of 9 meters. Finding Perimeters of Combination Figures When you are finding the perimeter of a combination figure, the perimeter is the distance around the outside of the figure and does not include any interior segments. Example Find the perimeter of each figure. A 7 in. in. Solution The tick marks indicate that the length of the rectangle is equal to the side length of the triangle. Find the sum of the 5 sides around the figure. P = = 5 The perimeter is 5 inches. 7 in. 7 in. in. in. 7 in. 35 UNIT 10 PERIMETER AND AREA
12 B m 5 m 3 m Solution Three sides of each square and one side of the triangle form the perimeter. P = = = 6 The perimeter is 6 meters. Application: Land Usage Example 5 Isaac is looking at a map that shows the boundaries of a city park in the shape of a quadrilateral. The lengths of three of the sides are 86 meters, 113 meters, and 9 meters. The length of the remaining side is smudged. Isaac calls the park s office and learns that the entire boundary of the park is 515 meters long. Find the length of the fourth side. Solution Write and solve an equation s = 515 The sum of the four side lengths equals the perimeter s = 515 Simplify on the left. s = Subtract 93 from both sides. The fourth side has a length of meters. Problem Set 1. Each side of a regular decagon is 15 centimeters long. Find the perimeter of the decagon.. Each side of a regular pentagon is inches long. Find the perimeter of the pentagon. 3. Find the perimeter of a square if a side length is 7 meters.. Find the perimeter of regular heptagon if a side length is 1 millimeters. PERIMETER 355
13 Find the perimeter of each figure km ft 6 km 7 ft mm cm m 1 m ft 3 in. 1 ft in. 7 6 in. 8 For problems 15 0, answer each question. 15. The perimeter of the rectangle is units. What is the width of the rectangle? 17. The perimeter of the triangle is 51 centimeters. What is the value of x? 16 x cm 1 cm 16. The perimeter of the figure is 19. meters. What is the length of each side of the figure? 16 cm 356 UNIT 10 PERIMETER AND AREA
14 18. The perimeter of a square is 8 yards. What is the length of each side of the square? 19. Find the length of a rectangle if its perimeter is 0 units and its width is 3 units. 0. The perimeter of a regular decagon is 161 centimeters. Find the length of each side of the decagon. Find the perimeter of each figure in.. 10 in. 10 mm 8 in. 3 mm 5 mm Write answers in complete sentences. 5. A farmer wants to build a fence around a grazing meadow. The meadow is shaped like a rectangle and is 110 meters long and 7 meters wide. How much fencing material must the farmer buy? 6. Kara glued 7 inches of yarn around a photo. What is the width of the photo if the length is 0 inches? 7. A sandbox is shaped like a regular hexagon with a side length of 8.5 meters. How far will a child walk if he walks along the entire border of the sandbox three times? * * 8. Find the perimeter of a rectangle whose vertices are located at (, 5), (1, 5), (1, 3), and (, 3). 9. Challenge Find the perimeter of a rectangle that has a width of 18 inches and a length of feet. 30. Challenge The length of a rectangle is twice its width. Find the width if the perimeter is 66 centimeters. PERIMETER 357
15 Areas of Rectangles and Triangles Every closed figure has an interior. The interior of the rectangle is the space enclosed by the sides of the rectangle. The interior of this rectangle is shaded. DEFINITION The area of a figure is the number of square units in the interior of the figure. This rectangle has an area of 3 square units. Notice that 3 is the product of the number of rows,, and number of columns, 8. TIP The interior is the inside of the figure. THINK ABOUT IT Area is expressed using square units, such as ft (square feet). When no units are provided, we use square units. Finding the Area of a Rectangle AREA OF A RECTANGLE The area of a rectangle with length l and width w is A = lw. l w Example 1 Find the area of the rectangle. 60 mm 11 mm Solution Use the formula. The calculation may be performed with or without the units. Method 1 Method A = lw A = lw = Substitute 60 for l and 11 for w. = (60 mm) (11 mm) = 660 Multiply. = mm mm = 660 mm The area is 660 square millimeters. TIP We generally use the first method because it is simpler. 358 UNIT 10 PERIMETER AND AREA
16 Finding the Area of a Triangle A triangle is half a rectangle, so the formula for the area of a triangle is half the formula for the rectangle. AREA OF A TRIANGLE The area of a triangle with base b and height h is A = 1 bh. h b The base of a triangle always forms a right angle with the height of the triangle. For acute triangles, the height is always shown inside the triangle. For obtuse triangles, it can be located in the exterior of the triangle. In a right triangle, the height can be one of the sides of the triangle. h b h b h b TIP Any side can be used as the base. The height will change accordingly. Example Find the area of the triangle. 5 km km Solution Use the formula. A = 1 bh = 1 5 Substitute for b and 5 for h. = 11 5 Multiply. = 75 Multiply. The area is 75 square kilometers. TIP The area can also be written as 75 km. AREAS OF RECTANGLES AND TRIANGLES 359
17 Finding Missing Lengths Example 3 A The area of the triangle is 5 square centimeters. What is the height of the triangle? 9 cm? Solution Substitute the known information into A = 1 bh. Solve for h. A = 1 bh Write the formula. 5 = 1 9 h Substitute 5 for A and 9 for b. 5 =.5h Simplify. 1 = h Divide both sides by.5. The height is 1 centimeters. B The area of a rectangle is 31 square inches. What is the length of the rectangle if the width is inches? Solution Substitute the known information into A = lw. Solve for l. A = lw Write the formula. 31 = l Substitute 31 for A and for w. 5.5 = l Divide both sides by. The length is 5.5 inches. Finding Areas of Combination Figures Example Find the area of the figure. 6 ft 9 ft 17 ft Solution Add the area of the triangle to the area of the rectangle. The base of the right triangle is the length of the rectangle, 17 ft. A = 1 bh + lw Use the formulas for areas of a triangle and a rectangle. = Substitute 17 for b and l, 6 for h, and 9 for w. = Multiply. = 0 Add. The area is 0 square feet. 360 UNIT 10 PERIMETER AND AREA
18 Finding the Difference of Areas Example 5 Find the area of the shaded region. 8 m m m Solution Subtract the area of the rectangle from the area of the square. A = s lw Use the formulas for area of a square and area of a rectangle. = 8 Substitute 8 for s, for l, and for w. = 6 8 Simplify. = 56 Subtract. The area of the shaded region is 56 square meters. Problem Set Find the area of each figure m 6. 1 km 8 m 1 km m 7 yd 1 m 3. 9 m 8 yd 1. m mm 13 mm 1 6 in. in ft 3 ft 18 cm 3.5 ft 3 cm AREAS OF RECTANGLES AND TRIANGLES 361
19 Answer each question. 11. The area of the rectangle is 11 square units. What is the width of the rectangle? 1. The area of the triangle is 5.5 square units. What is the base of the triangle? What is the length of a rectangle if its width is 7 meters and its area is 63 square meters? 15. What is the height of a triangle if its area is 19 square feet and its base is 16 feet? 16. The area of a square is 9 square meters. What is the length of each side of the square? 17. The area of a square is 9 square meters. What is the perimeter of the square? The area of the triangle is 38 square inches. What is the height of the triangle? 0 in. 3 in. Find the area of each figure yd 11 mm 7 mm 7 mm 3 mm mm 0. 5 ft 7 yd 5 yd 16 yd 5 ft 3 yd in. 18 ft 1 ft 6 in. 10 in. 8 in. 5 ft 36 UNIT 10 PERIMETER AND AREA
20 Find the area of the shaded region cm cm Write answers in complete sentences. 6. A basketball court is 9 feet long and 50 feet wide. What is the area of the basketball court? 7. Mr. Nunez has a back yard that is shaped like a right triangle with a base of 8 meters and a height of 60 meters. How much will it cost him to fertilize the yard if the cost is 3 cents per square meter? *8. Challenge Show that the area of the triangle is the same regardless of which side is used as the base. * * 9. Challenge Tell how the formula for the area of a triangle is related to the formula for the area of a rectangle. 30. Challenge Find the area of a triangle whose length is 1 centimeters and whose width is 55 millimeters AREAS OF RECTANGLES AND TRIANGLES 363
21 Special Quadrilaterals A quadrilateral can have zero pairs, one pair, or two pairs of parallel sides. TIP Arrows are used to indicate parallel lines and segments. DEFINITION A trapezoid is a quadrilateral with exactly one pair of parallel sides. DEFINITION A parallelogram is a quadrilateral with two pairs of parallel sides. Parallelograms are further classified by their side and angle measures. DEFINITION A rectangle is a quadrilateral with four right angles. DEFINITION A rhombus is a quadrilateral with four congruent sides. TIP The plural of rhombus is rhombi. 36 UNIT 10 PERIMETER AND AREA
22 DEFINITION A square is a quadrilateral with four congruent sides and four right angles. All rectangles, rhombi, and squares are also parallelograms. Classifying Quadrilaterals Example 1 For each figure, write all names that apply: trapezoid, parallelogram, rectangle, rhombus, and square. A Solution Both pairs of sides are parallel, so the figure is a parallelogram. Because all the angles are right angles, it is also a rectangle. B Solution Only one pair of sides is parallel. The figure is a trapezoid. C Solution All four angles are right angles and all four sides are congruent. The figure is a parallelogram, rectangle, rhombus, and square. SPECIAL QUADRILATERALS 365
23 Classifiying Quadrilaterals A chart can help you see how the special quadrilaterals are related. Quadrilaterals Trapezoids Parallelograms Rectangles Squares Rhombi THINK ABOUT IT A square can be defined as a rectangle with four congruent sides, or as a rhombus with four right angles. A figure always belongs to a classification above it in the chart, provided they are connected. For example, all parallelograms are quadrilaterals. A figure will sometimes belong to a classification below it, provided they are connected. For example, a quadrilateral is sometimes a parallelogram. Figures that are not connected will never belong to the same classification. For example, a parallelogram is never a trapezoid. Example Tell if each statement is always, sometimes, or never true. A A rectangle is a square. Solution Some rectangles are squares, but not all are. Only rectangles with all sides congruent are squares. The statement is sometimes true. B A square is a rectangle. Solution Every square is a rectangle because all squares have four right angles. The statement is always true. C A square is a parallelogram. Solution A square is always a parallelogram. Both pairs of sides are always parallel, so the statement is always true. D A rhombus is a trapezoid. Solution A rhombus always has two pairs of parallel sides, while a trapezoid always has exactly one pair of parallel sides. The statement is never true. 366 UNIT 10 PERIMETER AND AREA
24 Using Properties of Parallelograms Parallelograms have properties that other quadrilaterals do not have. PROPERTIES OF PARALLELOGRAMS The opposite sides of a parallelogram are congruent. The opposite angles of a parallelogram are congruent. TIP The opposite sides are the parallel sides. The opposite angles do not have a common side. Example 3 A AWRT is a parallelogram. Which sides are W R congruent? Which angles are congruent? Solution Opposite sides are congruent: AT WR and WA RT. Opposite angles are congruent: A R and W T. B Find the values of x and y in the parallelogram. B. cm 5.8 cm Solution Opposite sides of a parallelogram are congruent. BR KP, so x =.. BK RP, so y = 5.8. A R K T y cm x cm P Identifying Quadrilaterals on a Coordinate Grid Example The set of points (0, ), (3, 3), (0, ), ( 3, 3) identifies the vertices of a quadrilateral. Use the most specific description to tell which figure the points form. Solution y ( 3, 3) (0, ) (0, ) 1 (3, 3) 3 5 x Plot and connect the points. All sides are congruent. The quadrilateral is a rhombus. TIP You can measure to see that the sides have the same length. SPECIAL QUADRILATERALS 367
25 Problem Set For each figure, write all the names that apply: trapezoid, parallelogram, rectangle, rhombus, and square m m 6. m m in. 6 in. 6 in. 6 in. Tell if each statement is always, sometimes, or never true. 9. A square is a rhombus. 10. A rectangle is a trapezoid. 11. A quadrilateral is a square. 1. A square is a quadrilateral. Draw each figure. 16. A rectangle that is a rhombus. 17. A rectangle that is not a rhombus. 18. A parallelogram that is not a rectangle. 13. A rectangle is a rhombus. 1. A parallelogram is a square. 15. A rhombus is a parallelogram. 19. A parallelogram that is not a rhombus. 0. A trapezoid with two right angles. Each set of points identifies the vertices of a quadrilateral. Use the most specific description to tell which figure each set of points forms. 1. (, ), (1, ), (1, ), (, 0). (, ), (5, ), (5, 5), (, 5) 3. ( 6, ), ( 1, ), (, 6), ( 7, 6). (, ), (5, ), (5, 0), (, 0) 5. (0, 6), (3, 5), (0, ), ( 3, 5) 368 UNIT 10 PERIMETER AND AREA
26 Find the values of x and y in each parallelogram x 8 y x y 8. (y 6) 107 (x + 1) y 3x Answer the question. *30. Challenge Use a Venn diagram to illustrate the relationships among the quadrilaterals. SPECIAL QUADRILATERALS 369
27 Areas of Special Quadrilaterals The formulas for the area of a parallelogram and for a trapezoid are similar to the area formula for a rectangle. Parallelograms and trapezoids have bases and heights. A base is defined to be the bottom side of a geometric figure. The height is perpendicular to the base. It is the length of the segment that extends from the base to the opposite side. Finding the Area of a Parallelogram Every parallelogram has four bases; each side can be a base. The height depends on which side is used as the base. Heights are sometimes shown outside the parallelogram. b h b h h b THINK ABOUT IT Any side of a parallelogram can be the base because the parallelogram can be rotated so that any side is on the bottom. AREA OF A PARALLELOGRAM The area of a parallelogram with base b and height h is A = bh. h b THINK ABOUT IT When a parallelogram is a rectangle, the height is a side of the parallelogram and the terms length and width are used instead of base and height. Example 1 Find the area of the parallelogram. 3 in. 8 in. Solution A = bh Write the formula. = 3 8 Substitute 3 for b and 8 for h. = 7 Multiply. The area is 7 square inches. 370 UNIT 10 PERIMETER AND AREA
28 Finding the Area of a Trapezoid A trapezoid has two bases: b 1 and b. The parallel sides are always the bases. The height is the length of a segment that joins the bases and forms right angles with them. AREA OF A TRAPEZOID The area of a trapezoid with bases b 1 and b b 1 and height h is h A = 1 h(b + b ). 1 b Example Find the area of the trapezoid. 18 ft 7 ft 1 ft Solution A = 1 h(b + b ) Write the formula. 1 = 1 7 (18 + 1) Substitute 18 for b 1, 1 for b and 7 for h. THINK ABOUT IT It does not matter which base is used for b 1 and which is used for b. = Simplify inside the parentheses. = 105 Multiply. The area is 105 square feet. Finding Missing Lengths With a known area and some algebra, you can find missing side lengths. Example 3 A The area of a parallelogram is 675 square centimeters. What is the height of the parallelogram if its base is 5 centimeters long? Solution Substitute the known information into A = bh. Solve for h. A = bh Write the formula. 675 = 5 h Substitute 675 for A and 5 for b. 15 = h Divide both sides by 5. The height is 15 centimeters. (continued) AREAS OF SPECIAL QUADRILATERALS 371
29 B The area of the trapezoid is 5 meters. Find the unknown base length.? 6 m Solution 10 m A = 1 h(b + b ) Write the formula. 1 5 = 1 6 (b ) Substitute 5 for A, 6 for h, and 10 for one of the bases. 5 = 3 (b ) Multiply on the right. 18 = b Divide both sides by 3. 8 = b 1 Subtract 10 from both sides. The length of the unknown base is 8 meters. Check A = 1 h(b + b ) = (8 + 10) = 3 18 = 5 Application: Painting Example Each wall of a foursided garden shed is 10 feet long and 8 feet high and has one rhombusshaped window. The windows are congruent and each has a base of feet and a height of 1.5 feet. The gardener wants to paint the inside of the walls. A can of the paint covers about 350 square feet per gallon. How many cans of paint will she need for two coats? Solution Find the area to be painted. First, find the area that is covered with one coat. Subtract the area of the windows A = lw bh from the area of the walls. = Substitute values for the variables. = 30 1 Multiply. = 308 Subtract. She has to cover 308 square feet for one coat. Next, double that amount to find the area covered in two coats. 308 = 616 Multiply area of one coat by. Divide by 350 to find how many cans of paint she needs = 1.76 Divide by 350. The gardener needs cans of paint. THINK ABOUT IT You can also use A = (lw bh). 37 UNIT 10 PERIMETER AND AREA
30 Problem Set Find the area of each figure km km in. 1 in. 0 in.. 9 cm 30 in. 8 cm cm m 3 6 m 11 1 m yd yd 7 yd x mm y mm 1 ft 15 ft ft b a c 5 AREAS OF SPECIAL QUADRILATERALS 373
31 Answer each question. 13. The area of the parallelogram is 16 square units. What is the height of the parallelogram? 1. The area of the trapezoid is 56 square units. What is the height of the trapezoid?? How long is the base of a parallelogram if its area is 100 square meters and its height is 5 meters? 17. What is the height of a parallelogram whose base length is 16 meters and whose area is 136 square meters? 18. What is the height of a trapezoid whose bases have lengths of 9 centimeters and 1 centimeters and whose area is 5.5 square centimeters? 19. The area of a trapezoid is 65 square feet. The height is 10 feet and the length of one of the bases is 9 1 feet. Find the length of the other base The area of the trapezoid is 11 square yards. Find the unknown base length. 3 yd 1 yd? yd Find the area of each figure mm. 11 in. 6 in. 9 in. 9 in. 6 mm 6 mm 6 in. 11 in UNIT 10 PERIMETER AND AREA
32 Answer each question.. Find the area of a parallelogram whose vertices are located at ( 1, 1), (5, 1), (3, ), and ( 3, ). 5. Find the area of a trapezoid whose vertices are located at (0, ), (9, ), (5, ), and (, ). 6. Mia wants to apply two coats of paint to her deck. Her deck is shaped like a trapezoid with base lengths of 1 meters and 0 meters. The perpendicular distance between the bases measures 16 meters. If paint costs $ per gallon and one gallon of paint covers about 350 square meters, how much will it cost Mia to paint her deck? 7. Joey will both mow and rake a yard for a fee of $0.05/square meter. How much will Joey charge to mow and rake a front yard that is shaped like a trapezoid with bases of 30 meters and 35 meters and with a height of 8 meters? 8. Lee is making a rock garden in the shape of a rhombus. He wants the area of the garden to be exactly 50 square feet. Give two possible sets of dimensions Lee could use. *9. Challenge Find the area of a trapezoid whose base lengths are 1 foot and yards, and whose height is 18 inches. *30. Challenge Use diagrams to show why a rectangle with a length of 1 and a width of 6 has the same area as a parallelogram with a base length of 1 and a height of 6. AREAS OF SPECIAL QUADRILATERALS 375
33 Circumference The distance around a polygon is called its perimeter while the distance around a circle is called its circumference. DEFINITION The circumference of a circle is the distance around the circle. Finding the Circumference of a Circle Since ancient times, people have known that the ratio of the circumference to the diameter of any circle is a constant that is just a bit more than 3. This constant is called π (pi), which is a decimal number that never repeats and never ends. In calculations, it is often approximated as 3.1. CIRCUMFERENCE OF A CIRCLE The circumference of a circle with diameter d and radius r is C = πd or C = πr. d r THINK ABOUT IT In a given circle, the diameter is twice the radius, so C = πd = πr = πr. Answers that are found by substituting 3.1 for π are estimates and should include the approximately equal to ( ) symbol. Answers that use the symbol for π are exact answers. Example 1 Find the circumference of each circle. Give both exact and approximate answers. A circle A A 17 cm Solution Because the diameter is given, use C = πd. C = πd Write the formula. = π 17 Substitute 17 for d Substitute 3.1 for π. 53. Multiply. The circumference is exactly 17π centimeters or about 53. centimeters. TIP When using 3.1 for π, use three digits when writing the circumference. 376 UNIT 10 PERIMETER AND AREA
34 B circle with radius of 5 meters Solution Because the radius is given, use C = πr. C = πr Write the formula. = π 5 Substitute 5 for r. = 10π Multiply Substitute 3.1 for π. 31. Multiply. The circumference is exactly 10π meters or about 31. meters. TIP Find the exact answer in terms of π first, and then substitute a value of π to find an approximation. Finding Missing Lengths Example A The circumference of a circle is 18π feet. What is the radius? Solution Substitute the known information into C = πr. Solve for r. C = πr Write the formula. 18π = π r Substitute 18π for C. 18 = r Divide both sides by π. 9 = r Divide both sides by. The radius is 9 feet. B The circumference of a circle is 0 yards. What is the diameter? Solution Substitute the known information into C =πd. Solve for d. C =πd Write the formula. 0 =πd Substitute 0 for C. 0 π = d Divide both sides by π d Substitute 3.1 for π. 1.7 d Divide both sides by 3.1. The diameter is about 1.7 yards. Finding Perimeters of Partial and Combination Figures A semicircle is half a circle. To find the circumference of a semicircle, divide by : C = πd or C = 1 πr =πr. A quarter circle is onefourth of a 1 circle. To find the circumference of a quarter circle, divide by : C = πd 1 or C = πr =πr. (continued) CIRCUMFERENCE 377
35 Example 3 A Find the exact circumference of a semicircle with radius 5 centimeters. Solution Use the formula C = πr. C = πr Use the formula for circumference of a semicircle. C = π 5 Substitute 5 for r. C = 5π Simplify. The exact circumference is 5π centimeters. B Find the circumference of a quarter circle with diameter 6 inches. Use 3.1 to approximate π. Solution Use the formula C = πd. C = π 6 Subtitute 6 for d. C = 1.5π Simplify. C Substitute 3.1 for π. C.71 Multiply. The circumference is exactly 1.5π cm or about.71 cm. Example A The figure is made up of two semicircles and a rectangle. Find the perimeter of the figure. 6 in. Solution P = πd 1 + π d = π in. Add the circumference of the semicircles to the two sides of the rectangle. + π The diameters are 15 and 6. = 10.5π + 1 Simplify Substitute 3.1 for π. 5.0 Simplify. The perimeter is about 5 inches. 378 UNIT 10 PERIMETER AND AREA
36 B The figure is made up of two congruent squares and a quarter circle. Find the perimeter of the figure to the nearest tenth..5 in. Solution The side of each square is the radius of the quarter circle. P = 6s + πr = π.5 = π Simplify. Add the six sides of the square to the circumference of the quarter circle. Substitute.5 for s and r Substitute 3.1 for π Multiply Add. The perimeter is about 18.9 inches. Application: Sports Example 5 A bicycle wheel has a radius of 16 inches. It is rolled on the ground for one complete revolution. How far did the wheel travel? Solution The distance traveled equals the circumference of the wheel. C = πr Write the formula. = π 16 Substitute 16 for r. = 3π Multiply Substitute 3.1 for π. 100 Multiply. The wheel traveled about 100 inches. CIRCUMFERENCE 379
37 Problem Set For problems 1 8, the center of each circle is shown. Find the circumference of each circle. Give both exact and approximate answers cm 1 in ft m 5. y 8. y x x Answer each question. 9. The circumference of a circle is 19π inches. What is the radius of the circle? 10. The circumference of a circle is 76 centimeters. What is the diameter of the circle? The value of π can be approximated by 7. Estimate the circumference of each circle using m UNIT 10 PERIMETER AND AREA
38 Find the perimeter of each figure cm 5 m 1. 1 m 3 ft km Answer each question. 19. What is the circumference of a swimming pool if its diameter is 8.5 cm? 0. The bottom of a lamp shade has a circumference of about 60 inches. Estimate the diameter to the nearest tenth. 1. A ring has a diameter of 1.6 cm. Estimate the circumference of the ring.. A tire has a radius of 15 inches. How far does it travel in 5 revolutions? 3. Joseph is making a plant holder so that the pot sits partly above and partly below a wooden board. To cut the hole in the board, he needs to know the diameter of the circle, but because a plant is already in the pot, he cannot measure it directly. Instead, he measures how much string can be wrapped around the pot at the desired height. What will be the diameter of the circle he cuts in the board if he used 35 millimeters of string?. A pitcher s mound on a baseball field has a diameter of 18 feet. What is its circumference? 5. A gardener has 8 kilometers of fencing material. If she makes a circular garden and uses all her fencing material, what will be the radius of her garden? 6. At the center of a basketball court, the inner circle has a radius of feet and the outer circle has a radius of 6 feet. What is the difference in the circumferences of the circles? 7. Suri s ornament has a diameter of.75 inches and Ada s ornament has a diameter of 1.5 inches. How much greater is the circumference of Suri s ornament than Ada s ornament? *8. Challenge A wheel has a diameter of 1 inches. How many revolutions will it make after rolling 0 feet? Find the length of the darkened part of each circle. *9. Challenge. *30. Challenge. m 10 m 7 CIRCUMFERENCE 381
39 Areas of Circles In addition to its use in the formula for circumference, π can help you calculate the area of a circle. Finding the Area of a Circle AREA OF A CIRCLE The area of a circle with radius r is A = πr. d r TIP r is read r squared and means r r. Example 1 Find the area of each circle. Give both exact and approximate answers. A circle C 5 mm C Solution Use the formula with r = 5. A = πr Write the formula. = π 5 Substitute 5 for r. = 65π 5 = 5 5 = Substitute 3.1 for π Multiply. The area is exactly 65π square millimeters or about 1960 square millimeters. B circle D TIP In the area formula, the order of operations tells you that only the radius is squared. Do not square π. 8 cm D 38 UNIT 10 PERIMETER AND AREA
40 Solution The diameter is given. Divide to find the radius: 8 =. A = πr Write the formula. = π Substitute for r. = 16 π Simplify Substitute 3.1 for π. 50. Multiply. The area is exactly 16π square centimeters or about 50. square centimeters. Finding Missing Lengths Example A The area of a circle is 100π square meters. What is the radius? Solution Substitute the known information into A = πr. Solve for r. A = πr Write the formula. 100π = πr Substitute 100π for A. 100 = r Divide both sides by π. 10 = r Think: What number times itself is 100? The radius is 10 meters. B The area of a circle is 50 square inches. What is the diameter? Solution After solving for r, multiply by to find d. A = πr Write the formula. 50 = π r Substitute 50 for A. 50 π = r Divide both sides by π r Substitute 3.1 for π r Divide. r Think: 16 =. The diameter is about 8 inches. Finding Areas of Partial and Combination Figures To find the area of a semicircle, divide by : A = πr. To find the area of a quarter circle, divide by : A = πr. (continued) AREAS OF CIRCLES 383
41 Example 3 The radius of the semicircle and height of the triangle are shown. Find the area of the figure. Solution A = πr + 1 bh Add the area of the semicircle to the area of the triangle. = π The base of the triangle is =. = 60.5π + 0 Simplify Substitute 3.1 for π Multiply. 10 Add. The area is about 10 square units Application: Food Example A small pizza has a diameter of 10 inches, a medium pizza has a diameter of 13 inches, and a large pizza has a diameter of 16 inches. A Estimate the difference in the areas of a medium and large pizza. Solution Find the area of each pizza. Medium: A = πr Large: A = πr = π 6.5 = π 8 = π.5 = π Subtract to find the difference: = 68. The difference is about 68 square inches. B Angie ate onefourth of a small pizza. About how many square inches of pizza did she eat? Solution Find the area of a quarter circle with a radius of 5 inches. A = πr = π 5 Write the formula. Substitute 5 for r. = 6.5π Simplify Substitute 3.1 for π Multiply. Angie ate about 19.6 square inches of pizza. REMEMBER Divide each diameter by to find each radius. 38 UNIT 10 PERIMETER AND AREA
42 C A pizza with a 1inch diameter costs $1.95 while a 1inch pizza costs $ Which pizza is a better deal? Solution Find the unit price of each pizza by dividing the cost of the pizza by the area. 1 inch diameter 1 inch diameter A = πr A = πr = π 7 = π Unit price $1.95 Unit price $ in 113 in $0.08 per square inch $0.097 per square inch The 1inch pizza is the better deal. Finding Areas by Subtraction Example 5 Find the area of the shaded region. 8 m 16 m Solution A = lw πr Subtract the area of the circle from the area of the rectangle. = 16 8 π Substitute 16 for l, 8 for w, and for r. = 18 16π Simplify Substitute 3.1 for π Multiply Subtract. The area of the shaded region is about 77.8 square feet. AREAS OF CIRCLES 385
43 Problem Set The center of each circle is shown. Find the area of each circle. Give both exact and approximate answers cm 1 mm ft m y x y x 5. y x y x Answer each question. 11. The area of a circle is 16π square meters. What is the radius of the circle? 1. The area of a circle is 36π square feet. What is the diameter of the circle? 13. The area of a circle is 1.5 square centimeters. What is the diameter of the circle? 1. The area of a circle is 15 square millimeters. What is the radius of the circle? 386 UNIT 10 PERIMETER AND AREA
44 Find the area of each figure in. 5 m 1 m 3 ft cm Find the area of the shaded region ft 3. 1 km 16 ft.. 18 mm 3 Answer each question. 5. What is the area of a swimming pool if its diameter is 1 meters? 6. A pizza with a 1inch diameter costs $1.99 while a 1inch pizza costs $9.99. Which pizza is a better deal? 7. An 18inch pizza costs $.99 while a 16inch pizza costs $ Which pizza is a better deal? 8. The center circle on a soccer field has an area of 100π square meters. Find the circumference of the center circle. 9. Which area is greater: a circle with a diameter of 10 kilometers or a square with a side length of 10 kilometers? 30. Ms. Brady s old waffle maker made circular waffles with a diameter of 17 centimeters. Her new waffle maker makes rectangular waffles that are centimeters long and 13 centimeters wide. Which makes waffles with a greater area? How much greater? *31. Challenge Explain how you would find the area of a figure with this shape. AREAS OF CIRCLES 387
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