Circumference and 7-1 Area of Circles MAIN IDEA Find the circumference Measure and record the distance d across the circular and area of circles. part of an object, such as a battery or a can, through its center. New Vocabulary circle Place the object on a piece of paper. Mark the point center where the object touches the paper on both the object radius and on the paper. chord diameter Carefully roll the object so that it makes one complete circumference rotation. Then mark the paper again. pi Finally, measure the distance C between the marks. Math Online glencoe.com • Extra Examples • Personal Tutor • Self-Check Quiz in. 1234 56 1. What distance does C represent? C 2. Find the ratio _ for this object. d 3. Repeat the steps above for at least two other circular objects and compare the ratios of C to d. What do you observe? 4. Graph the data you collected as ordered pairs, (d, C). Then describe the graph. A circle is a set of points in a plane center radius circumference that are the same distance from a (r) (C) given point in the plane, called the center. The segment from the center diameter to any point on the circle is called (d) the radius. A chord is any segment with both endpoints on the circle. The diameter of a circle is twice its radius or d 2r. A diameter is a chord that passes = through the center. It is the longest chord. The distance around the circle is called the circumference. The ratio of the circumference of a circle to its diameter is always 3.1415926…. 22 It is represented by the Greek letter π (pi). The numbers 3.14 and _ C 7 are often used as approximations for π. So, _ = π. This can also be d written as C = πd or C ≈ 3.14d. 352 Chapter 7 Measurement: Area and Volume 352_357_C07_L01_874050.indd 352 9/22/07 12:37:24 PM Gr8 MS Math SE ©09 - 874050 Circumference of a Circle Key Concept Words The circumference C of a circle Model C is equal to its diameter d times π, d r or 2 times its radius r times π. Symbols C = πd or C = 2πr Calculating with π Find the Circumferences of Circles When evaluating expressions involving π, Find the circumference of each circle. Round to the nearest tenth. use the π key on a calculator to obtain the 1 C = πd Circumference of a circle most accurate result. 9 in. However, using 3.14 for π C = π · 9 Replace d with 9. will result in a close approximation. C = 9π This is the exact circumference. Use a calculator to find 9π. 9 [π] 28.27433388 The circumference is about 28.3 inches. 2 C = 2πr Circumference of a circle 7.2 cm C = 2 · π · 7.2 Replace r with 7.2. C ≈ 45.2 Use a calculator. The circumference is about 45.2 centimeters. a. b. c. 7.5 in. 6 ft 11.7 m A circle can be decomposed into congruent wedge-like pieces. Then the pieces can be rearranged to form a figure that resembles a parallelogram. 1 C 2 radius 1 C 2 Since the circle has an area that is relatively close to the area of the parallelogram-shaped figure, you can use the formula for the area of a parallelogram to find the formula for the area of a circle. A = bh Area of a parallelogram 1 The base of the parallelogram is one-half the A = _ · C r (2 ) circumference and the height is the radius. 1 A = _ · 2πr r Replace C with 2πr. (2 ) A = π · r · r or πr 2 Simplify. Lesson 7-1 Circumference and Area of Circles 353 352_357_C07_L01_874050.indd 353 9/22/07 12:37:31 PM Gr8 MS Math SE ©09 - 874050 Area of a Circle Key Concept Words The area A of a circle is equal to Model π times the square of the radius r. r Symbols A = π r 2 Find the Areas of Circles Estimation Find the area of each circle. Round to the nearest tenth. To estimate the area of a 2 circle, square the radius 3 A = πr Area of a circle and then multiply by 3. 8 km A = π · 8 2 Replace r with 8. A = π · 64 Evaluate 8 2 . This is the exact area. A ≈ 201.1 Use a calculator. The area is about 201.1 square kilometers. 2 4 A = πr Area of a circle 15 ft A = π (7.5) 2 Replace r with half of 15 or 7.5. A = π · 56.25 Evaluate 7 .5 2 . This is the exact area. A ≈ 176.7 Use a calculator. The area is about 176.7 square feet. Find the area of each circle. Round to the nearest tenth. d. The radius is 11 inches. e. The diameter is 5 meters. 5 STATE PARKS Suppose you walk around the edge of the circular Point State Park fountain and estimate its circumference to be 470 feet. Based on your estimate, what is the approximate diameter of the fountain? C = πd Circumference of a circle 470 = πd Replace C with 470. _470 π = d Divide each side by π. 149.6 ≈ d Use a calculator. The diameter of the fountain is about 150 feet. f. HOME DECOR A catalog states that a circular area rug covers 19.5 square feet. What is the approximate diameter of the rug? 354 Chapter 7 Measurement: Area and Volume 352_357_C07_L01_874050.indd 354 9/22/07 12:37:32 PM Gr8 MS Math SE ©09 - 874050 Find the circumference of each circle. Round to the nearest tenth. Examples 1, 2 1. 2. 3. (p. 353) 18 cm 12 yd 2.5 mi Find the area of each circle. Round to the nearest tenth. Examples 3, 4 4. 5. 6. (p. 354) 14.5 m 21 ft 18.25 in. Example 5 7. BRACELETS When Cammie finished making a friendship bracelet, it was (p. 354) 7.9 inches long. What was the diameter of the bracelet? HOMEWORK HELP Find the circumference of each circle. Round to the nearest tenth. For See 8. 9. 10. 11. Exercises Examples 24 mm 38 mi 10 in. 8–11 1, 2 17 km 12–15 3, 4 16–19 5 Find the area of each circle. Round to the nearest tenth. 12. 13. 14. 15. 19.4 m 5.3 mi 7.25 ft 4.75 in. 16. PETS Simone purchased a circular exercise pen with a radius of 2.5 feet to keep her new puppy safe. Find the area inside the pen. 1 17. MEASUREMENT A circular table top has a radius of 2 _ feet. A decorative trim 4 is placed along the outside edge of the table. How long is the trim? 18. SAFETY A light in a parking lot illuminates a circular area 15 meters across. What is the area of the parking lot covered by the light? 19. BICYCLES Jerrod’s mountain bike has a tire diameter of 26 inches. How far will the bike travel in 100 rotations of its tires? Find the exact circumference and area of each circle. 20. The radius is 3.5 centimeters. 21. The diameter is 8.6 kilometers. 22. The diameter is 9 inches. 23. The radius is 0.6 mile. Lesson 7-1 Circumference and Area of Circles 355 352_357_C07_L01_874050.indd 355 9/22/07 12:37:35 PM Gr8 MS Math SE ©09 - 874050 24. Find the diameter of a circle if its area is 706.9 square millimeters. 25. GARDENING Mr. Townes created a 2-foot wide garden path around a circular garden. The radius of the garden is 7 feet. He wants to cover the path in stones. If Mr. 7 ft 2 ft Townes needs 1 bag of stones for every 5 square feet of path, how many bags of stones will he need to cover his garden path? 22 Another approximate value for π is _ . Use this value to find the 7 circumference and area of each circle. 1 26. The diameter is 7 feet. 27. The radius is 2 _ inches. 3 28. BAKING Joaquim is baking giant cookies for the school bake sale. They will be sold for $20 for one large cookie or $20 for three smaller 12 in. 8 in. cookies. Which offer is the better buy? Explain your reasoning. -BSHF 4NBMM 29. SPORTS Three tennis balls are packaged one on top of the other in a can. Which measure is greater, the can’s height or circumference? Explain. 30. TREES During a construction project, barriers are placed around trees. For each inch of trunk diameter, the protection 1 zone should have a radius of 1_ feet. 2 Find the area of this zone for a tree with 1 a trunk circumference of 63 inches. d in. 12 d ft 31. GRAPHIC ARTS Michael is painting a sign for a new coffee shop. On the sign, he drew a circle with a radius of 2 feet. He then drew another circle with a radius 1.5 times larger. How much greater is the area of the larger circle? 32. FIND THE DATA Refer to the Data File on pages 16–19. Choose some data EXTRA PRACTICE and write a real-world problem in which you would determine the area of See pages 685, 706. a circle. H.O.T. Problems 33. OPEN ENDED Draw and label a circle that has a circumference between 5 and 10 centimeters.