2 Cut the circle along the fold lines to divide the circle into 16 equal wedges. radius. Think About It

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Activity 8.7 Finding Aea of Cicles Question How do you find the aea of a cicle using the adius?

2 Cut the cicle along the fold lines to divide the cicle into 16 equal wedges. 3 Aange the wedges to fom a shape esembling a paallelogam.

The aea of the paallelogam in Step 3 can be used to appoximate the aea of the oiginal cicle. Wite a vebal expession fo the aea of the paallelogam. 2.

1 Activity 8.7 Finding Aea of Cicles Question How do you find the aea of a cicle using the adius? Mateials compass staightedge scissos Exploe 1 Use a compass to daw a cicle on a piece of pape. Cut the cicle out. Fold the cicle in half, fou times. 2 Cut the cicle along the fold lines to divide the cicle into 16 equal wedges. 3 Aange the wedges to fom a shape esembling a paallelogam. The base and height of the paallelogam ae labeled. adius adius half the cicumfeence Think About It Student Help VOCABULARY TIP The cicumfeence is the distance aound a cicle. 1. The aea of the paallelogam in Step 3 can be used to appoximate the aea of the oiginal cicle. Wite a vebal expession fo the aea of the paallelogam. 2. The atio of the cicumfeence C to the diamete d of a cicle is denoted by the Geek lette π (o pi), which is appoximately equal to So, C πd. Because the diamete is twice the adius of the cicle, C πd π(2) 2π. Use the fact that C 2π to wite you vebal expession fom Execise 1 using vaiables. Simplify the expession. 3. Find the aea of the cicle at the ight. Use 3.14 as an appoximation fo π. 4 Activity 451

2 8.7 Cicumfeence and Aea of Cicles Goal Find the cicumfeence and aea of cicles. Key Wods cicle cente adius diamete cicumfeence cental angle secto A cicle is the set of all points in a plane that ae the same distance fom a given point, called the cente of the cicle. A cicle with cente P is called cicle P, o P. The distance fom the cente to a point on the cicle is the adius. The plual of adius is adii. The distance acoss the cicle, though the cente, is the diamete. The diamete d is twice the adius. So, d 2. The cicumfeence of a cicle is the distance aound the cicle. adius diamete cicumfeence Fo any cicle, the atio of the cicumfeence to its diamete is denoted by the Geek lette π, o pi. The numbe π is , which is an iational numbe. This means that π neithe teminates no epeats. So, an appoximation of 3.14 is used fo π. cente CIRCUMFERENCE OF A CIRCLE Wods Cicumfeence π(diamete) 2π(adius) Symbols C πd o C 2π EXAMPLE 1 Find the Cicumfeence of a Cicle Find the cicumfeence of the cicle. 4 in. Student Help STUDY TIP When simplifying an expession involving π, substitute 3.14 fo π. You can also use the π key on you calculato, as in Examples 2 and 3. C 2π Fomula fo the cicumfeence 2π(4) Substitute 4 fo. 8π Simplify. 8(3.14) Use 3.14 as an appoximation fo π Multiply. ANSWER The cicumfeence is about 25 inches. 452 Chapte 8 Polygons and Aea

3 Find the Cicumfeence of a Cicle Find the cicumfeence of the cicle. Round you answe to the neaest whole numbe cm 9 ft 16 in. AREA OF A CIRCLE Wods Aea π(adius) 2 Symbols A π 2 EXAMPLE 2 Find the Aea of a Cicle Find the aea of the cicle. 7 cm Student Help KEYSTROKE HELP If you calculato has π witten above a key, use the following keystokes to simplify 49π: 49 A π 2 Fomula fo the aea of a cicle π(7) 2 Substitute 7 fo. 49π Simplify Use a calculato. ANSWER The aea is about 154 squae centimetes. EXAMPLE 3 Use the Aea of a Cicle Find the adius of a cicle with an aea of 380 squae feet. A π 2 Fomula fo the aea of a cicle 380 π 2 Substitute 380 fo A Divide each side by π. Use a calculato. 11 Take the positive squae oot. A 380 ft 2 ANSWER The adius is about 11 feet. 8.7 Cicumfeence and Aea of Cicles 453

4 Find the Aea of a Cicle Find the aea of the cicle. Round you answe to the neaest whole numbe in. 3 cm 12 ft Student Help VOCABULARY TIP The tem adius is also used to name a segment that connects the cente of a cicle to a point on the cicle. Two such adii ae used to detemine a secto of a cicle. Cental Angles An angle whose vetex is the cente of a cicle is a cental angle of the cicle. A egion of a cicle detemined by two adii and a pat of the cicle is called a secto of the cicle. Because a secto is a potion of a cicle, the following popotion can be used to find the aea of a secto. Aea of secto Aea of entie cicle Measue of cental angle Measue of entie cicle secto cental angle EXAMPLE 4 Find the Aea of a Secto Find the aea of the blue secto. 9 m Fist find the aea of the cicle. A π 2 π(9) The aea of the cicle is about 254 squae metes. 2 Then find the aea of the secto. Let x equal the aea of the secto. Visualize It! A cicle contains two staight angles. So, thee ae , o 360, in a cicle o 360 Aea of secto Aea of entie cicle x x 254 p x 30,480 Measue of cental angle Measue of entie cicle Substitute. Coss poduct popety Simplify x 0 30, Divide each side by x Simplify. ANSWER The aea of the secto is about 85 squae metes. 454 Chapte 8 Polygons and Aea

5 Find the Aea of a Secto In Execises 7 and 8, A epesents the aea of the entie cicle and x epesents the aea of the blue secto. Complete the popotion used to find x. Do not solve the popotion. 7. A 22 m 2 8. A 28 ft 2 x? x??? Find the aea of the blue secto. Round you answe to the neaest whole numbe cm 2 ft in Execises Guided Pactice Vocabulay Check 1. Sketch a cicle. Sketch and label a adius and a diamete of the cicle. 2. Descibe how to find the cicumfeence of a cicle given its adius. Skill Check Copy and complete the table below Radius, Diamete, d? 14 in. 11 cm? 3.5 m?? 1 ft d Wite an equation fo the aea A o the cicumfeence C by filling in the missing numbe. 7. C 2π? 8. A? (3) 2 9. A π(? ) Cicumfeence and Aea of Cicles 455

Pactice and Applications Exta Pactice See p. 690. Finding Cicumfeence Find the cicumfeence of the cicle. Round you answe to the neaest whole numbe. 10. 11. 12. 7 cm 10 m 2 in. 13.

6 Pactice and Applications Exta Pactice See p Finding Cicumfeence Find the cicumfeence of the cicle. Round you answe to the neaest whole numbe cm 10 m 2 in. 13. A cicle with a adius of 13 yads 14. A cicle with a diamete of 15 metes Finding Aea Find the aea of the cicle. Round you answe to the neaest whole numbe ft 1 cm 6 in m 16 ft 19 yd 21. Canbeies To havest canbeies, the field is flooded so that the beies float. What aea of canbeies can be gatheed into a cicula egion with a adius of 5.5 metes? 22. Eo Analysis A student was asked to find the aea of the cicle below. Descibe any eos. A π 2 (3.14)(12) Homewok Help Example 1: Exs Example 2: Exs Example 3: Exs Example 4: Exs You be the Judge 23. One of you classmates states that if the adius of a cicle is doubled, then its aea doubles. Do you agee o disagee? Explain you easoning using the cicles at the ight Chapte 8 Polygons and Aea

IStudent Help I CLASSZONE.COM HOMEWORK HELP Exta help with poblem solving in Exs. 24 26 is at classzone.com Using Algeba Use the aea A of the cicle to find the adius.

7 IStudent Help I CLASSZONE.COM HOMEWORK HELP Exta help with poblem solving in Exs is at classzone.com Using Algeba Use the aea A of the cicle to find the adius. Round you answe to the neaest whole numbe. 24. A 13 in A 531 m A 154 ft A cicle has an aea of 50 squae units. What is its adius? 28. A cicle has an aea of 452 squae units. What is its diamete? 29. A cicle has an aea of 28 squae units. What is its diamete? 30. Challenge The cicle at the ight has an aea of 78.5 squae yads. What is its cicumfeence? A 78.5 yd 2 Caees Sectos and Popotions In Execises 31 and 32, A epesents the aea of the entie cicle and x epesents the aea of the blue secto. Complete the popotion used to find x. Do not solve the popotion. 31. A 12 ft A 198 yd 2 x?? x 360? ? Aea of Sectos Find the aea of the blue secto. Stat by finding the aea of the cicle. Round you answe to the neaest whole numbe m 6 ft cm LANDSCAPE ARCHITECTS use engineeing knowledge and ceativity to design natual envionments, such as paks, gadens, and eceational aeas. Caee Links CLASSZONE.COM Landscaping The diagam shows the aea of a lawn coveed by a wate spinkle. Round you answe to the neaest whole numbe. 36. What is the aea of the lawn that is coveed by the spinkle? 37. Suppose the wate pessue is weakened so that the adius is 12 feet. What is the aea of lawn that will be coveed? 8.7 Cicumfeence and Aea of Cicles 457

8 EXAMPLE Find Complex Aeas Find the aea of the shaded egion. 1 Fist find the aea of the oute egion, which is a squae. Aea of squae s 2 (12) ft 2 12 ft 2 Then find the aea of the inne egion, which is a cicle. The diamete of the cicle is 12 feet, so the adius is 6 feet. Aea of cicle π 2 π(6) ft 2 3 Finally, subtact the aea of the cicle fom the aea of the squae. Aea of shaded egion Aea of squae Aea of cicle ANSWER The aea of the shaded egion is about 31 squae feet. Finding Complex Aeas Use the method in the example above to find the aea of the shaded egion. Round you answe to the neaest whole numbe cm 6 cm 2 m 2 m 10 cm 8 m Standadized Test Pactice 40. Multi-Step Poblem Eath has a adius of about 3960 miles at the equato. Suppose a cable is wapped aound Eath at the equato, as shown in the diagam. a. Find the length of the cable by finding the cicumfeence of Eath. (Assume that Eath is pefectly ound. Use 3.14 fo π.) b. Suppose 10 miles is added to the 3960 mi cable length in pat (a). Use this length as the cicumfeence of a new cicle. Find the adius of the lage cicle. (Use 3.14 fo π.) c. Suppose you ae standing at the equato. How fa ae you fom cable the cable in pat (b)? 458 Chapte 8 Polygons and Aea

9 Mixed Review Identifying Quadilateals Use the infomation in the diagam to name the special quadilateal. (Lesson 6.6) Squaes and Rectangles Find the aea of the polygon. (Lesson 8.3) ft 14 in. 11 ft 6 m 2 m Algeba Skills Integes Evaluate the expession. (Skills Review, p. 663) (5) 49. (4)(11) (12) (3) (3) (5)(13) (9) Quiz 3 Find the aea of the figue. In Execises 5 and 6, ound you answe to the neaest whole numbes. (Lessons ) ft 6.1 cm 5.2 cm 9 ft 3 m 3 m 4. 5 yd mm 4 yd 4 in. 4 yd In Execises 7 9, A gives the aea of the figue. Find the missing measue. In Execise 9, ound you answe to the neaest whole numbe. (Lessons ) 7. A 195 cm 2 8. A 57 ft 2 9. A 201 in. 2 h 6 ft 7 ft 15 cm b Cicumfeence and Aea of Cicles 459

8.7 Circumference and Area

8.7 Circumference and Area Page 1 of 8 8.7 Cicumfeence and Aea of Cicles Goal Find the cicumfeence and aea of cicles. Key Wods cicle cente adius diamete cicumfeence cental angle secto A cicle is the set of all points in a plane