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


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1 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, and are in the shape of geometric figures such as triangles or parallelograms. You will calculate the area enclosed by a typical regatta course..1 Weaving a Rug Area and Perimeter of Rectangles and Squares p. 1.4 Signs, Signs, Every Place There Are Signs! Area of Regular Polygons p Boundary Lines Area of Parallelograms and Triangles p Say Cheese! Area and Circumference of a Circle p The Keystone Effect Area of a Trapezoid p Installing Carpeting and Tile Area and Perimeter of Composite Figures p. 191 Chapter Perimeter and Area 11
2 Introductory Problem for Chapter Fencing for a Garden A neighbor fenced in her backyard and offers you her leftover fencing. You want to use the fencing to surround a vegetable garden in your yard. The diagram shown is a layout of your property, including measurements and all buildings. Currently, your property has no fencing. Describe the shape, location, and dimensions of your garden that maximize the area, given the following lengths of fencing feet of fencing 2. 0 feet of fencing. 50 feet of fencing Be prepared to share your methods and solutions. 12 Chapter Perimeter and Area
3 .1 Weaving a Rug Area and Perimeter of Rectangles and Squares OBJECTIVES In this lesson you will: l Calculate the areas of rectangles. l Calculate the perimeters of rectangles. l Calculate the areas of squares. l Calculate the perimeters of squares. l Determine the effect of altering the dimensions of a rectangle or square on the perimeter or area. l Construct a square and a rectangle. Bhadra is an artist who creates specialty rugs. She creates rugs on a machine called a loom. On the loom, a piece of thread called a weft thread is woven through vertical pieces of thread called warp threads. Lesson.1 Weaving a Rug 1
4 PROBLEM 1 A Rectangular Rug Bhadra is currently creating rectangularshaped rugs. Take Note A rectangle is a quadrilateral that has four right angles. 1. One rectangular rug is seven feet long and three feet wide. Draw a model of this rug on the grid shown. Each square on the grid represents a square that is one foot long and one foot wide. 2. What is the area of this rug? Explain your reasoning.. What is the perimeter of this rug? Explain your reasoning. 14 Chapter Perimeter and Area
5 4. Draw six different rectangles on the grid shown. Use the letters A through F to name each rectangle. Each square on the grid represents a square that is one foot long and one foot wide. 5. Complete the table to show the length, width, area, and perimeter of each rectangle. Rectangle A B C D E F Length (units) Width (units) Perimeter (units) Area (square units) 6. What is an example of two rectangles having the same area, but different dimensions? a. What are the perimeters of these rectangles? b. If the areas are equal, are the perimeters always equal? Lesson.1 Weaving a Rug 15
6 7. Can you determine the perimeter of a rectangle without drawing it if you know the rectangle s length and width? If so, explain how you can do this. 8. Write a formula that you can use to calculate the perimeter of any rectangle. Use for the length of the rectangle, w for the width of the rectangle, and P for the perimeter. Make sure that the expression for the perimeter is in simplest form. 9. Can you determine the area of a rectangle without drawing it if you know the rectangle s length and width? If so, explain how you can do this. 10. Write a formula that you can use to calculate the area of any rectangle. Use for the length of the rectangle, w for the width of the rectangle, and A for the area of the rectangle. 11. Can you determine the area of a rectangle if its perimeter is known? Explain. 12. Can you determine the perimeter of a rectangle if its area is known? Explain. 16 Chapter Perimeter and Area
7 1. For each rectangle, either the length, width, or area of the rectangle is unknown. First calculate the value of the unknown measure. Then calculate the perimeter. a. 15 feet 21 feet b. Area: 48 square millimeters 8 millimeters c. Area: square inches.5 inches Lesson.1 Weaving a Rug 17
8 14. Calculate the perimeter and area of a rectangle that is 11 meters long and 5 meters wide. a. Double the length and width of the rectangle. Calculate the perimeter of the new rectangle. b. What effect does doubling the length and width have on the perimeter? c. Do you think that doubling the length and width will have the same effect on the area? 15. Calculate the area of the rectangle that is 22 meters long and 10 meters wide. 16. What effect does doubling the length and width have on the area? 18 Chapter Perimeter and Area
9 PROBLEM 2 Square Rugs Bhadra has also received several requests to create squareshaped rugs. Take Note A square is a quadrilateral that has four right angles and four congruent sides. A square can also be considered a rectangle with four congruent sides. 1. One square rug is seven feet long and seven feet wide. Draw a model of this rug on the grid shown. Each square on the grid represents a square that is one foot long and one foot wide. 2. What is the area of this rug? Explain.. What is the perimeter of this rug? Explain. Lesson.1 Weaving a Rug 19
10 4. Draw six different squares on the grid. Use the letters A through F to name each square. Each square on the grid represents a square that is one foot long and one foot wide. 5. Complete the table to show the length, width, area, and perimeter of each square. Square A B C D E F Length (units) Width (units) Perimeter (units) Area (square units) 140 Chapter Perimeter and Area
11 6. Can you determine the perimeter of a square without drawing the square if you know the length of one side of the square? Explain. 7. Write a formula that you can use to calculate the perimeter of any square. Use s for the side length of the square and P for the perimeter. 8. Can you determine the area of a square without drawing it if you know the length of one side of the square? Explain. 9. Write a formula that you can use to calculate the area of any square. Use s for the length of a side of the square and A for the area of the square. Lesson.1 Weaving a Rug 141
12 10. Calculate the value of the unknown side length, area, and perimeter in the squares shown. a. 5 centimeters b. Area: 169 square feet 142 Chapter Perimeter and Area
13 11. Calculate the perimeter and area of a square that has a side length equal to 9 inches. a. Double the side length of the square. Calculate the perimeter of the new square. b. What effect does doubling the side length of a square have on the perimeter? c. Do you think that doubling the side length of a square will have the same effect on the area? Explain. 12. Calculate the area of a square that has the side length equal to 10 meters. a. Double the length of the side of the square. Calculate the area of the new square. b. What effect does doubling the length of a side of the square have on the area? Lesson.1 Weaving a Rug 14
14 PROBLEM Construction A Square Given the Perimeter 1. The perimeter of a square is shown. A B a. Write a paragraph to explain how you will construct this square. b. Construct the square 144 Chapter Perimeter and Area
15 PROBLEM 4 Construction A Rectangle Given the Perimeter 1. The perimeter of a rectangle is shown. A B a. Write a paragraph to explain how you will construct this rectangle that is not a square. b. Construct the rectangle that is not a square. Be prepared to share your methods and solutions. Lesson.1 Weaving a Rug 145
16 146 Chapter Perimeter and Area
17 .2 Boundary Lines Area of Parallelograms and Triangles OBJECTIVES In this lesson you will: l Calculate the areas of parallelograms. l Calculate the areas of triangles. l Calculate the perimeters of triangles. l Solve for unknown measures of triangles. l Construct an isosceles triangle. KEY TERMS l parallelogram l altitude of a parallelogram l height of a parallelogram l altitude of a triangle l height of a triangle PROBLEM 1 A Parallelogram Rug Bhadra has a special request from a client. The client would like a rug in the shape of a parallelogram. A model of the rug is shown on the grid. Each square on the grid represents a square that is one foot long and one foot wide. Take Note A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Lesson.2 Boundary Lines 147
18 1. Explain how you can create a rectangle from the parallelogram shown so that the rectangle and the parallelogram have the same area. Then, check your answer by demonstrating your method on a separate sheet of grid paper. Draw your rectangle on top of the parallelogram. 2. What is the area of the rectangle from Question 1? Explain.. What is the area of the rug? Explain. 148 Chapter Perimeter and Area
19 4. Bhadra s client requests another parallelogramshaped rug. A model of the new rug is shown on the grid. Calculate the area of the rug. Explain your reasoning. Each square on the grid represents a square that is one foot long and one foot wide. Any side of a parallelogram is a base. EFGH is the same parallelogram drawn in different orientations. Each square on the grid represents a square that is one foot long and one foot wide. F G E E base F H H base G Lesson.2 Boundary Lines 149
20 An altitude of a parallelogram is a line segment drawn from a vertex, perpendicular to the line containing the opposite side. A height of a parallelogram is the perpendicular distance from any point on one side to the line containing the opposite side. F G altitude E base H E F altitude H base G 5. For each parallelogram, draw a segment that represents a height. Label the height with its measure and label the base with its measure. Each square on the grid represents a square that is one foot long and one foot wide. 150 Chapter Perimeter and Area
21 6. Write a formula for the area of a parallelogram. Use b for the base of the parallelogram, h for the height, and A for the area. 7. For each parallelogram, either the length, width, or area of the parallelogram is unknown. Calculate the value of the unknown measure. a. 9.5 feet 10 feet b. Area: 60 square meters 15 meters c. Area: 28 square inches.5 inches Lesson.2 Boundary Lines 151
22 8. Bhadra charges $20 per square foot of rug for a basic design. A client orders one basic rectangularshaped rug that is 6 feet long and 4 feet wide and one basic parallelogramshaped rug with a base that is 8 feet long and a height that is feet. What is the total cost for the rugs? Explain your reasoning. PROBLEM 2 The Race Course One of the typical shapes of a sailboat race course is triangularshaped. The course path is identified by buoys called marks. When the course is triangularshaped, the marks are located at the vertices of the triangle. A sample course with the marks numbered is shown. Each square on the grid represents a square that is one tenth of a kilometer long and one tenth of a kilometer wide. 2 1 Wind 1. How many grid squares in a row create an area that is one kilometer long and one tenth of a kilometer wide? 152 Chapter Perimeter and Area
23 2. How many grid squares are in an area that is one kilometer long and one kilometer wide?. Estimate the area enclosed by the course. Explain. 4. Is your area from Question exact? 5. On the previous grid, use two sides of the triangle to draw a parallelogram. 6. Calculate the area of the parallelogram. 7. Can you calculate the exact area of the triangle by using the area of the parallelogram? Why or why not? Lesson.2 Boundary Lines 15
24 8. Calculate the exact area enclosed by the triangular course. 9. How does the exact area enclosed by the triangularshaped course compare to the estimate? 10. How does the area of the parallelogram relate to the area of the triangle? 11. Consider the race course shown on the grid. Each square on the grid represents a square that is one tenth of a kilometer long and one tenth of a kilometer wide. Calculate the area enclosed by the course Chapter Perimeter and Area
25 12. What information about the triangle did you need to calculate the area in Question 11? Any side of a triangle is a base. Triangle KYM is the same triangle drawn in three different orientations. K Y M Y base M M base K K base Y An altitude of a triangle is a line segment drawn from a vertex perpendicular to a line containing the opposite side. A height of a triangle is the perpendicular distance from a vertex to the line containing the base. K Y M altitude Y base altitude M M base K K base Y altitude 1. Write a formula that you can use to calculate the area of any triangle. Use b for the length of the base, h for the height, and A for the area of the triangle. Lesson.2 Boundary Lines 155
26 14. Determine the base and height of triangles KYM, MYK, KMY. Then calculate the area of each triangle. K Y M Y base M M base K K base Y 15. Describe what happens to the height of a triangle as the length of the base changes when the area remains the same. 16. Calculate the unknown measure in each triangle. a. Area: 60 square meters 24 meters b. 6 yards 8 yards 156 Chapter Perimeter and Area
27 c. Area: 42 square inches 6 inches 17. The original race course is shown, but now the lengths of the legs of the race are given. If a boat must complete the course once, how long is the race? kilometers kilometers 1.6 kilometers Wind 18. What geometric name is given to this measurement? Lesson.2 Boundary Lines 157
28 19. In sailboat races, it is common for a boat to have to go around a course more than once, or revisit a leg of the course more than once. Suppose that to complete the race, a boat must sail to the marks in the following order: 1, 2,, 1,, 1, 2,, 1,. How long is this race? 20. If a boat is competing in a race, do you think that the boat will travel more than or less than the race length you calculated in Question 19. PROBLEM Boundary Lines 1. Several triangles are drawn on the grid shown. Given: KN PR K M N a. Calculate the area of triangle KPR. P R b. Calculate the area of triangle MPR. c. Calculate the area of triangle NPR. 158 Chapter Perimeter and Area
29 d. Compare the areas of triangles KPR, MPR, and NPR. e. Compare the bases of triangles KPR, MPR, and NPR. f. Compare the heights of triangles KPR, MPR, and NPR. g. What conclusion can be made about triangles that share the same base, or have bases of equal measure, and also have equal heights? 2. Use the conclusion you made in Question 1, part (g) to solve this problem. A sister and brother inherit equal amounts of property; however, the boundary line separating their land is not straight. Your job is to draw a new boundary line that is straight and keeps the property division equal. Explain how you solved the problem. A Upper boundary line Sister s Land Brother s Land B Lower boundary line Lesson.2 Boundary Lines 159
30 PROBLEM 4 Construction Isosceles Triangle Given the Perimeter 1. Construct an isosceles triangle with the given perimeter. A B 2. Did all of your classmates construct the same isosceles triangle? Explain. Be prepared to share your methods and solutions. 160 Chapter Perimeter and Area
31 . The Keystone Effect Area of a Trapezoid OBJECTIVES In this lesson you will: l Calculate the areas of trapezoids. l Calculate unknown measures of trapezoids. l Construct an isosceles trapezoid. KEY TERMS l bases of a trapezoid l legs of a trapezoid l altitude of a trapezoid l height of a trapezoid In most classrooms, a projection screen is hung above the blackboard. Generally, the screen is located higher than where the projector sits. To view images on the screen, the projector must be tilted upward. This tilting can cause keystoning, which is a distortion of the image. A normal image and possible distorted image are shown. Things to remember when interviewing for a job Dress neatly Arrive early Be polite PROBLEM 1How Distorted? 1. Describe how the normal image has been distorted. 2. Describe the shapes formed by the normal image and the distorted image. Lesson. The Keystone Effect 161
32 . Which image do you think has a larger area? Explain. 4. The normal image and the distorted image are shown on the grid. Each square on the grid represents a square that is four inches long and four inches wide. Calculate the area of each image and write it in the center of the image. 5. How do the areas of the images compare? 6. Is your area of the distorted image exact? 7. Consider the distorted image in Question 4. How can you use the area formulas you already know to calculate the exact area of this image? 162 Chapter Perimeter and Area
33 8. Calculate the exact area of the distorted image. 9. How do the exact areas of the normal image and the distorted image compare? 10. Consider the distorted image again. Suppose that you make an exact copy of this image, flip it vertically, and move it next to the image as shown. a. What is the geometric figure that is formed from these images? Lesson. The Keystone Effect 16
34 b. Calculate the area of the geometric figure in Question 1. Then, calculate the area of the distorted image. 11. Was it easier to calculate the area of the distorted image by using your method in Question 7 or by using the method in Question 10? Explain. The distorted image is a trapezoid. The parallel sides of the trapezoid are called the bases of the trapezoid. Nonparallel sides are the legs of the trapezoid. Trapezoid TRAP is the same trapezoid drawn in different orientations. P base A T base R Take Note A trapezoid is a quadrilateral with exactly one pair of parallel sides. T base R P base A P A T R base base base base R T A P 164 Chapter Perimeter and Area
35 An altitude of a trapezoid is a line segment drawn from a vertex perpendicular to a line containing the opposite side. A height of a trapezoid is the perpendicular distance from a vertex to the line containing the base. P base A T base R altitude altitude T R P A P A T R altitude base base altitude R A P T 12. Consider the trapezoid shown on the left. Suppose that you make an exact copy of this trapezoid, flip it vertically, and move it next to the trapezoid as shown. Label the bases of the trapezoid on the right. h b 1 b 2 a. Write a formula for the area of the entire figure. b. Write a formula for the area of one of the trapezoids. Explain. Lesson. The Keystone Effect 165
36 1. For each trapezoid, either a height, the length of one base, or the area is unknown. Determine the value of the unknown measure. a. 22 millimeters 8 millimeters 6 millimeters b. feet Area: 25 square feet 7 feet c. 6 meters 9 meters Area: 45 square meters 166 Chapter Perimeter and Area
37 14. The projector in Problem 1 was tilted differently to create the distorted image shown. Each square on the grid represents a square that is four inches long and four inches wide. Normal Image Distorted Image a. What is the area of the distorted image? b. How does the area of the distorted image compare to the area of the normal image? Lesson. The Keystone Effect 167
38 PROBLEM 2 Construction A Trapezoid 1. Construct a trapezoid, given AB is the perimeter. A B 2. Did all of your classmates construct the same trapezoid? Explain. 168 Chapter Perimeter and Area
39 PROBLEM Summary Area Formulas l Area of a Rectangle: A w w l Area of a Square A s 2 S S l Area of a Parallelogram: A bh h b l Area of a Triangle: A 1 2 bh l Area of a Trapezoid: A 1 2 (b 1 b 2 )h h b 1 b h b 2 Lesson. The Keystone Effect 169
40 170 Chapter Perimeter and Area
41 .4 Signs, Signs, Every Place There Are Signs! Area of Regular Polygons OBJECTIVE In this lesson you will: l Calculate the areas of regular polygons. KEY TERMS l congruent polygons l apothem Have you ever noticed that every stop sign looks exactly the same, every yield sign looks exactly the same, and so on? This is because the Federal Highway Administration has standards that indicate the exact sizes and colors of roadway signs. Most of the sign shapes are polygons. PROBLEM 1 How Big is that Sign? The specifications for the smallest possible yield sign are shown. 0 in. YIELD 60 o 60 o 0 in. 0 in. 60 o 1. What is special about the triangle that forms the yield sign? Lesson.4 Signs, Signs, Every Place There Are Signs! 171
42 The specifications for a Do Not Enter sign are shown. 0 in. DO NOT 90 o 90 o ENTER 0 in. 0 in. 90 o 90 o 0 in. 2. What is special about the quadrilateral that forms the Do Not Enter sign? The specifications for a stop sign are shown. STOP 12.4 in in. 15 o 15 o 12.4 in. 15 o 15 o 12.4 in in. 15 o 12.4 in. 15 o 15 o 15 o 12.4 in in.. What is special about the octagon that forms the stop sign? The polygons in Questions 1 through are special polygons called regular polygons. Two other possible sizes for a yield sign are shown. Take Note A regular polygon is a polygon with all sides congruent and all angles congruent. 6 in. 60 o 60 o 6 in. 6 in. 60 o 48 in. 60 o 60 o 48 in. 48 in. 60 o 172 Chapter Perimeter and Area
43 4. Are these signs regular polygons? What can you conclude about all regular triangles? 5. The yield sign from Question 1 is shown with its approximate height. Calculate the approximate area of the yield sign. 0 in. YIELD 26 in. 6. Calculate the approximate area of the Do Not Enter sign from Question 2. When two polygons are exactly the same size and exactly the same shape, the polygons are said to be congruent polygons. Lesson.4 Signs, Signs, Every Place There Are Signs! 17
44 7. To calculate the area of the stop sign from Question, you can use the fact that a regular polygon can be divided into triangles that are all exactly the same size and same shape. The bases of the triangles are the sides of the polygon as shown. In this case, the height of each triangle is approximately 15 inches. Calculate the area of the stop sign. Round your answer to the nearest tenth if necessary. Explain your reasoning in. 15 in. The height of the triangle in the stop sign in Question 7 is the apothem of the octagon. The apothem of a regular polygon is the perpendicular distance from the center of the regular polygon to a side of the regular polygon. 8. Draw a segment that represents an apothem on each regular polygon shown. The center of the polygon is marked by a point. 174 Chapter Perimeter and Area
45 9. The hexagon shown is a regular hexagon. Calculate the area of the hexagon. Explain your reasoning. 40 cm 4.6 cm 10. The heptagon shown is a regular heptagon. Calculate the area of the heptagon. Explain your reasoning. 8 m 8. m 11. Explain how you can calculate the area of a regular polygon if you know the length of the apothem and the length of each side. Lesson.4 Signs, Signs, Every Place There Are Signs! 175
46 12. Write a formula for the area of a regular polygon with n sides. Use a for the length of the apothem and for the length of one side of the polygon. PROBLEM 2 Who's Correct? 1. Lily claims the formula for determining the area of a regular polygon is A ( 1 2 a ) n, where is the length of a side, a is the apothem, and n is the number of sides. Molly claims the formula for determining the area of a regular polygon is A 1 Pa, where P is the perimeter of the polygon and a is the apothem. 2 Who is correct? Explain. 2. Emma thinks the definition for a regular polygon is too long and it should be shortened. She believes that if a polygon has all sides equal in length, then all angles will always be equal in measure. a. What are two examples Emma could use to justify her conclusion? 176 Chapter Perimeter and Area
47 b. Is Emma correct? Justify your conclusion.. Jath also thinks the definition for a regular polygon is too long. He states that if a polygon has all angles of equal measure, then all sides will always be equal in length. a. What are two examples Jath could use to justify his conclusion? b. Is Jath correct? Justify your conclusion. Lesson.4 Signs, Signs, Every Place There Are Signs! 177
48 PROBLEM Calculate the Area 1. The length of one side of a regular nonagon is 24 feet and the length of the apothem is approximately feet. Calculate the area of the regular nonagon. 2. The side length of the largest possible stop sign is 20 inches and the length of the apothem is approximately 24.1 inches. a. What is the area of the largest possible stop sign? b. The side length of the smallest possible stop sign is 9.9 inches and the length of the apothem is approximately 12 inches. What is the area of the smallest possible stop sign? c. How many times larger is the area of the largest possible stop sign than the area of the smallest possible stop sign? Explain. Be prepared to share your methods and solutions. 178 Chapter Perimeter and Area
49 .5 Say Cheese! Area and Circumference of a Circle OBJECTIVES In this lesson you will: l Calculate the circumferences of circles. l Calculate the areas of circles. l Calculate the areas of squares. l Calculate unknown measures of circles. KEY TERMS l circle l diameter l radius l circumference l irrational number l concentric circles l annulus Did you know your eyes and a camera lens have a lot in common? The amount of light that enters a camera lens is controlled by the aperture of the lens. Your pupil functions as the aperture of your eye. The aperture of a lens has a circular shape, and its size can change like the size of your pupil changes. A circle is the set of all points in a plane equidistant from a given point, which is the center of the circle. The circumference of a circle is the distance around the circle. The diameter of a circle is the distance across the circle through the circle s center. The radius of a circle is one half the diameter. The plural form of radius is radii. Lesson.5 Say Cheese! 179
50 PROBLEM 1 The Distance Around the Aperture 1. One size of an aperture for a camera lens is shown on the grid. Each square on the grid represents a square that is 5 millimeters long and 5 millimeters wide. a. Calculate the diameter of the aperture. b. Calculate the circumference of the circle. Explain your reasoning. c. What is the ratio of the circumference to the diameter? d. Write this ratio as a decimal. Round your answer to two decimal places if necessary. 2. Another size of a camera lens aperture is shown on the grid. Each square on the grid represents a square that is 5 millimeters long and 5 millimeters wide. 180 Chapter Perimeter and Area
51 a. Calculate the diameter and circumference of the aperture. b. Write the ratio of the circumference to the diameter as a decimal. Round your answer to two decimal places if necessary.. Another size of a camera lens aperture is shown. Each square on the grid represents a square that is 5 millimeters long and 5 millimeters wide. a. Calculate the diameter and circumference of the aperture. b. Write the ratio of the circumference to the diameter as a decimal. Round your answer to two decimal places if necessary. 4. How do the ratios in Questions 1 through compare to each other? It turns out that the ratio of a circle s circumference to its diameter is the same for all circles. The value of this ratio is an irrational number, its decimal representation neither repeats nor terminates. Use the symbol (read as pi ) to represent the exact value of this ratio. 5. Write an equation that defines as a ratio of measures of a circle. In this course, it is useful to use the fraction 22, or the decimal.14 to 7 approximate. Lesson.5 Say Cheese! 181
52 6. Write a formula that you can use to calculate the circumference of a circle when you know the diameter. Use C for the circumference and use d for the diameter. 7. Now write a formula that you can use to calculate the circumference of a circle when you know the radius. Use C for the circumference and use r for the radius. 8. Use either of your formulas in Question 6 or 7 to calculate the circumferences of the apertures in Questions 1 through. Use.14 for. 9. Suppose that the radius of an aperture is 5 millimeters. a. What is the circumference of the aperture? Leave your answers in terms of. b. Suppose that the radius is increased by one millimeter. Do you think that the circumference of the aperture will be increased by one millimeter? If so, justify your answer. If not, how does the circumference change? c. Suppose that the radius is doubled to 10 millimeters. Do you think that the circumference of the aperture will be doubled? If so, justify your answer. If not, how does the circumference change? Explain. 182 Chapter Perimeter and Area
53 PROBLEM 2 The Area Inside the Aperture 1. Consider the aperture shown. Each square on the grid represents a square that is 5 millimeters long and 5 millimeters wide. a. What is the diameter of the aperture? What is the radius? b. Consider the shaded square in Question 1. How does the side length of the square relate to the circle? c. What is the area of the shaded square? d. Estimate the number of shaded squares it will take to cover the entire circle. e. What is the estimated area of the aperture? Lesson.5 Say Cheese! 18
54 2. Another camera lens aperture is shown. Each square on the grid represents a square that is 5 millimeters long and 5 millimeters wide. Use the method shown in Question 1 to estimate the area of the aperture.. Another camera lens aperture is shown. Each square on the grid represents a square that is 10 millimeters long and 10 millimeters wide. Estimate the area of the aperture. 184 Chapter Perimeter and Area
55 4. Another camera lens aperture is shown. Each square on the grid represents a square that is 10 millimeters long and 10 millimeters wide. Estimate the area of the aperture. 5. Complete the table that shows the results from Questions 1 through 4. Radius (millimeters) Area of Shaded Square (square millimeters) Number of Shaded Squares Needed to Cover Circle Estimated Area of Circle 6. What do you notice about the number of shaded squares needed to cover each circle in the table? 7. For each circle, what is the relationship between the area of the radius square and the estimated area of the circle? Lesson.5 Say Cheese! 185
56 8. Is the number of shaded squares close to a number that you know? If so, name the number. The area A of a circle with a radius r is given by A r Use the formula to calculate the areas of the apertures described in Question 5. Use.14 for. 10. In each of the following circles you know one measure. Calculate the unknown measures: the radius, diameter, area, and/or circumference. Leave your answers in terms of. a. 8 inches 186 Chapter Perimeter and Area
57 b. Area: 144 square inches c. Circumference: 10 centimeters Lesson.5 Say Cheese! 187
58 PROBLEM Area of Circles 1. Eight diameters divide a circle into 16 equal parts as shown in Figure 1. Each of the 16 parts is drawn separately in Figure 2. All parts are placed together flipping every other piece vertically in Figure. Figure 1 Figure 2 Figure a. What geometric shape does Figure most closely resemble? b. Represent the approximate base and height of Figure in terms of the radius and circumference of the circle. c. Calculate the area of Figure. How does the area of Figure compare to the area of the circle? 2. The radius of a circle is 1 cm. a. Calculate the area of the circle. b. Suppose the radius is doubled. What effect will this have on the area? 188 Chapter Perimeter and Area
59 PROBLEM 4 Concentric Circles Concentric circles are circles that share the same center. 1. Use your compass to draw 2. Shade the region in between or two concentric circles. bounded by the concentric circles. An annulus is the region bounded by two concentric circles.. Calculate the area of the annulus drawn if R, the radius of the larger circle, is equal to 8 cm and r, the radius of the smaller circle, is equal to cm. r R Be prepared to share your methods and solutions. Lesson.5 Say Cheese! 189
60 190 Chapter Perimeter and Area
61 .6 Installing Carpeting and Tile Area and Perimeter of Composite Figures OBJECTIVE In this lesson you will: l Calculate the areas of composite figures. KEY TERM l composite figure Most companies that do home renovations and repairs routinely send out a person to record measurements needed for the job. These measurements are necessary to estimate the cost of the job, and to order the materials to complete the job. PROBLEM 1 A Brand New Floor A carpeting company has been hired to install flooring on the first floor of a home. A diagram of this first floor is shown. 5 feet 10 feet 5 feet Kitchen Dining room 8 feet 10 feet Enclosed porch 14 feet 10 feet Living room 12 feet 1. Calculate the unknown lengths a and b. b a Lesson.6 Installing Carpeting and Tile 191
62 2. The home owners would like to install indoor/outdoor carpeting on the enclosed porch. How many square feet of indoor/outdoor carpeting will be needed?. The home owners would like to install loop carpeting in the living room. How many square feet of loop carpeting will be needed? 4. The home owners would like to install tile in the kitchen. How many square feet of tile will be needed? 5. The home owners would like to install wood flooring in the dining room. How many square feet of wood flooring will be needed? 192 Chapter Perimeter and Area
63 6. What is the total area of the first floor? Explain your reasoning. 7. An employee from the carpeting company must now calculate the total cost of the materials for the job. Wood flooring and tile are sold by the square foot, but carpeting is sold by the square yard. The employee must rewrite some of his areas in square yards. a. Draw a square that models one square foot. Use one inch to represent one foot. b. There are three feet in one yard. Draw a square that models one square yard. Use one inch to represent one foot. Lesson.6 Installing Carpeting and Tile 19
64 c. Divide the square you drew for Question 7(b) into square feet. How many square feet are there in one square yard? Explain your reasoning. d. What is the area of the enclosed porch in square yards? e. What is the area of the living room in square yards? Round your answer to the nearest tenth. f. The indoor/outdoor carpeting costs $10.75 per square yard, the loop carpeting costs $7.50 per square yard, the wood flooring costs $4.50 per square foot, and the tile costs $4.25 per square foot. Calculate the total cost of the materials needed for the job. 194 Chapter Perimeter and Area
65 g. Do you think that the total cost is accurate? Why or why not? h. After the home owners saw the total cost of the flooring for the job, they decided that the wood flooring was too expensive and decided to buy the same loop carpeting in the dining room as in the living room. Calculate the total cost of the flooring for the first floor if the Harris family decides to buy the loop carpeting for the dining room. Lesson.6 Installing Carpeting and Tile 195
66 PROBLEM 2 Composite Figures The floor plan used in Problem 1 can also be referred to as a composite figure, meaning a figure formed by adjoining different shapes. 1. Calculate the area of each figure. Use.14 for. a. 6 feet 6 feet b. 8 yards yards 8 yards yards 2. Calculate the area of the shaded portion of each figure. Use.14 for. a. 20 centimeters 0 centimeters 196 Chapter Perimeter and Area
67 b. 16 meters 40 meters c. 10 inches 40 inches d. A circle inscribed in a square. 15 m Lesson.6 Installing Carpeting and Tile 197
68 e. Two small circles tangent to each other and tangent to the large circle. 8 in. 8 in. f. One medium and one small circle tangent to each other and tangent to the large circle. m 12 m 198 Chapter Perimeter and Area
69 g. A rectangle inscribed in a circle. 6 cm 10 cm 8 cm h. A circle inside a regular hexagon. 2 in. 2 in. 6 in. Be prepared to share your methods and solutions. Lesson.6 Installing Carpeting and Tile 199
70 200 Chapter Perimeter and Area
71 Chapter Checklist KEY TERMS l rectangle (.1) l legs of a trapezoid (.) l square (.1) l altitude of a trapezoid (.) l parallelogram (.2) l height of a trapezoid (.) l altitude of a parallelogram (.2) l isosceles trapezoid (.) l height of a parallelogram (.2) l congruent polygons (.4) l altitude of a triangle (.2) l apothem (.4) l height of a triangle (.2) l circle (.5) l bases of a trapezoid (.) l diameter (.5) l radius (.5) l circumference (.5) l irrational number (.5) l concentric circles (.5) l annulus (.5) l composite figure (.6) CONSTRUCTIONS l square given perimeter (.1) l rectangle given perimeter (.1) l isosceles triangle given perimeter (.2) l trapezoid (.).1 Calculating Perimeters and Areas of Rectangles To calculate the perimeter of a rectangle, multiply the length by 2 and the width by 2, and then calculate the sum of the products. To calculate the area of a rectangle, multiply the length by the width. Example: P 2 2w 2(6) 2(4) 20 feet A w 6(4) 24 square feet 6 ft 4 ft The perimeter of the rectangle is 20 feet, and the area of the rectangle is 24 square feet. Chapter Checklist 201
72 .1 Calculating Perimeters and Areas of Squares To calculate the perimeter of a square, multiply the length of a side by 4. To calculate the area of a square, multiply the length of a side by itself. Example: 12 m 12 m P 4s 4(12) 48 meters A s s s square meters The perimeter of the square is 48 meters, and the area of the square is 144 square meters..2 Calculating Areas of Parallelograms To calculate the area of a parallelogram, multiply the base by the height. Example: 10.5 m 22 m A bh 22(10.5) 21 square meters The area of the parallelogram is 21 square meters. 202 Chapter Perimeter and Area
73 .2 Calculating Areas of Triangles To calculate the area of a triangle, multiply one half of the base by the height. Example: 7 cm 11 cm. A 1 2 bh 1 2 (11)(7) 8.5 square centimeters The area of the triangle is 8.5 square centimeters. Calculating Areas of Trapezoids To calculate the area of a trapezoid, multiply one half of the sum of the bases by the height. Example: 5 yd 0 yd A 1 2 (b b )h (5 65)(0) square yards 65 yd The area of the trapezoid is 1500 square yards. Chapter Checklist 20
74 .4 Calculating Areas of Regular Polygons To calculate the area of a regular polygon, multiply one half of the length of one side of the polygon times the length of the apothem times the number of sides. Example: 7 in. 4.8 in..5 A ( 1 2 a ) n 1 2 (7)(4.8)(5) 84 square inches The area of the regular pentagon is 84 square inches. Calculating Circumferences and Areas of Circles To calculate the circumference of a circle, multiply the diameter by. You can also multiply the radius by 2. To calculate the area of a circle, multiply the square of the radius by. You can use.14 to approximate. Example:.25 ft C d 2 r A r 2 2(.14)(.25) (.14)(.25 2 ) feet.17 square feet The circumference of the circle is approximately feet and the area of the circle is approximately.17 square feet. 204 Chapter Perimeter and Area
75 .6 Calculating Areas of Composite Figures To calculate the area of a composite figure, calculate the area of each common figure that makes up the composite figure and then calculate the sum of the areas. Example You have a flower bed in a corner of your backyard. A diagram of the flower bed is shown. You want to cover the flower bed with mulch. What is the area that you will be covering with mulch? Area of square (4)(4) 16 square feet Area of rectangle (7)(4) 28 square feet Area of quarter of circle 1 4 ( )(42 ) 1 (.14)(16) square feet 4 Area of composite figure square feet You will be covering an area of about square feet with mulch. Chapter Checklist 205
76 206 Chapter Perimeter and Area
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