11-4 Area of Circles MAIN IDEA Find the areas of circles. • Fold a paper plate in half four times to New Vocabulary • Label the radius r as shown. Let C sector Math Online glencoe.com • Extra Examples • Personal Tutor • Self-Check Quiz divide it into 16 equal-sized sections. 1 C 2 r represent the circumference of the circle. • Cut out each section; reassemble to form a parallelogram-shaped figure. 1. What is the measurement _ of the base and the height? 1 C ; r 2 r (height) 2. Substitute these values into 1 C (base) 2 the formula for the area of a parallelogram. A = 1 C (r ) _ 2 3. Replace C with the expression for the circumference of a circle, 2π r. Simplify the equation and describe what it represents. _ 3. A = 1 (2πr) (r); A = πr 2; the area of a circle 2 In the Mini Lab, the formula for the area of a parallelogram was used to develop a formula for the area of a circle. Area of a Circle Words Key Concept The area A of a circle equals the product of π and the square of its radius r. Symbols Model r A = πr 2 Find the Area of a Circle 1 Find the area of the circle. A = πr 2 Area of a circle A = π · 22 Replace r with 2. [π] 2 2 in. 12.56637061 The area of the circle is approximately 12.6 square inches. a. Find the area of a circle with a radius of 3.2 centimeters. Round to the nearest tenth. 32.2 cm 2 Lesson 11-4 Area of Circles 589_0589_0593_CH11_L4_874046 589 589 9/26/07 7:34:46 PM 2 COINS Find the area of the face of the Wisconsin quarter shown. The diameter of the quarter is 24 millimeters, 1( ) so the radius is _ 24 or 12 millimeters. 24 mm 2 A = πr 2 Area of a circle A = π · 12 2 Replace r with 12. A ≈ 452.4 Use a calculator. The area is approximately 452.4 square millimeters. b. POOLS The bottom of a circular swimming pool with diameter 30 feet is painted blue. How many square feet are blue? 706.5 ft 2 Radii The plural form of radius is radii. A sector of a circle is a region of a circle bounded by two radii. 3 Ellis draws a circle with a diameter of 16 inches, and shades one region of the circle. Find the approximate area of the sector. 16 in. A 100 in 2 C 402 in 2 180° B 201 in 2 D 804 in 2 Read the Item Identifying What is Given Before finding area, be sure to read the question carefully and identify if the radius or diameter is given. The diameter of the circle is 16 inches. Since there are 360° in 1 the area of the entire circle. 180° a circle, the sector is _ or _ 360° 2 Solve the Item A = πr 2 A=π·8 Area of a circle 2 A ≈ 200 Replace r with 16 ÷ 2 or 8. Multiply. Use 3.14 for π. 1 The area of the sector is approximately _ (200) or 100 square inches. 2 The answer is A. c. Ray drew one circle with a radius of 7 centimeters and another circle with a radius of 10 centimeters. Find the approximate difference between the areas of the circles. H F 28 cm 2 590 G 40 cm 2 H 160 cm 2 J 254 cm 2 Chapter 11 Measurement: Two- and Three-Dimensional Figures 590_0589_0593_CH11_L4_874046 590 9/26/07 7:34:52 PM ★ indicates multi-step problem Examples 1, 2 Find the area of each circle. Round to the nearest tenth. (pp. 589–590) 78.5 cm 1. 5 cm 2 254.5 in 2. 2 4. diameter = 13 ft 9 in. 132.7 ft 2 Example 3 5. MULTIPLE CHOICE Kenneth draws the circle (p. 590) shown at the right. He shades one region of the circle. What is the approximate area of the sector? B A 88 yd 2 201.1 m 2 3. diameter = 16 m 14 yd C 310 yd 2 B 154 yd 2 D 615 yd 2 Answers were computed using the π key on a calculator. HOMEWORK For Exercises 6–7, 10–11, 14–15, 19 8–9, 12–13, 16–18 36–38 HELP See Examples Find the area of each circle. Round to the nearest tenth. 201.1 cm 2 7. 6. 28.3 in 2 3 in. 95.0 ft 2 8. 11 ft 1 8 cm 2 3 Exercise Levels A: 6–19 B: 20–29 C: 30–35 227.0 cm 2 10. 9. 17 cm 12. diameter = 8.4 m 55.4 m 2 15. radius = 3_ ft 2 3 4 2.4 m 18.1 m 2 3.2 mm 13. diameter = 12.6 cm 124.7 cm 2 16. diameter = 9_ mi 2 32.2 mm 2 11. 1 4 14. radius = 4_ in. 1 2 63.6 in 2 17. diameter = 20_ yd 44.2 ft 67.2 mi 18. PATCHES Find the area of the Girl Scout patch shown if the diameter is 1.25 inches. Round to the nearest tenth. 1.2 in 2 338.2 yd 2 3 4 19. TOOLS A sprinkler that sprays water in a circular area can be adjusted to spray up to 30 feet. To the nearest tenth, what is the maximum area of lawn that can be watered by the sprinkler? 2,827.4 ft 2 ESTIMATION Estimate to find the approximate area of each circle. 20. 8 cm Sample answer: 3 × 42 = 48 cm 2 21. 5.9 ft Sample 22. answer: 3 × 62 = 108 ft 2 13.8 in. Sample answer: 3 × 72 = 147 in 2 Lesson 11-4 Area of Circles 591_0589_0593_CH11_L4_874046 591 591 9/26/07 7:34:55 PM For Exercises 23–26, use a compass to draw the circle shown on centimeter grid paper. 23. Count the number of squares that lie completely within the circle. Then count the number of squares that lie completely within or contain the circle. 24. Estimate the area of the circle by finding the mean of the two values you found in Exercise 23. 25. Find the area of the circle by using the area formula. 26. How do the areas you found in Exercises 24 and 25 compare to one another? 27. A semicircle is half a circle. Find the area of the semicircle to the nearest tenth. 8.6 m 28. Which has a greater area, a triangle with a base of 100 feet and a height of 100 feet or a circle with diameter of 100 feet? Justify your selection. EXTRA PRACTICE 29. RADIO SIGNALS A radio station sends a signal in a circular area with an 80- mile radius. Find the approximate area in square kilometers that receives the signal. (Hint: 1 square mile ≈ 2.6 square kilometers) See pages 698, 714. H.O.T. Problems 30. REASONING If the length of the radius of a circle is tripled, does the area also triple? Explain your reasoning. 4 ft 12 ft CHALLENGE Find the area of the shaded region in each figure. Round to the nearest tenth. 31. 32. 33. 8m 3.5 cm 12 m 5.25 in. 12.5 cm 34. FIND THE ERROR Dasan and Carmen are finding the area of a circle that has a diameter of 16 centimeters. Who is correct? Explain. A = π(16)2 ≈ 804 cm2 A = π(8)2 ≈ 201 cm2 Dasan 35. Carmen WR ITING IN MATH Write and solve a real-world problem in which you would solve the problem by finding the area of a circle. 592 Chapter 11 Measurement: Two- and Three-Dimensional Figures 589_593_C11_L4_892329.indd 592 4/8/10 1:59:58 PM 36. The radius of the half dollar in 38. Which two figures have the same area shaded? A centimeters is given below. Find the approximate area of the shaded sector. A 8m 1.95 cm 7.5 m 12 m 12 m Figure II Figure I 10 m 12 m 12 m 180° 6m A 6 cm 2 C 14 cm 2 B 12 cm 2 D 28 cm 2 A Figure I and Figure IV 37. Which equation could be used to find the area in square inches of a circle with a radius of 12 inches? J F A=6×π B Figure I and Figure II C Figure II and Figure IV D Figure II and Figure III H A = 12 × π G A = π × 62 J Figure IV Figure III A = π × 12 2 39. MEASUREMENT What is the circumference of a circle that has a radius of 8 yards? Use 3.14 for π and round to the nearest tenth if necessary. (Lesson 11-3) 50.2 yd 40. MEASUREMENT Find the area of a triangle with a base of 21 meters and a height of 27 meters. (Lesson 11-2) 283.5 m 2 Find the area of each parallelogram. Round to the nearest tenth if necessary. (Lesson 11-1) 41. 39.5 cm 2 42. 5 cm 10 in. 120 in 2 12 in. 44. 8.5 2 72.25 45. 3.14 · 6 2 113.04 46. 8.7 m 7.9 cm PREREQUISITE SKILL Simplify each expression. 100.1 m 2 43. 11.5 m (Lessons 1-2, 1-3, and 1-4) _1 · 5.4 2 + 11 25.58 47. _1 · 7 2 + (9)(14) 150.5 2 2 Lesson 11-4 Area of Circles 593_0589_0593_CH11_L4_874046 593 593 9/26/07 7:35:14 PM

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