5.10. Area and Perimeter INSERT

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Curtis Moore
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1 5.10 Aea and Peimete INSERT A Peimete We egin tis section y eviewing te definition of a polygon, and te definition of peimete. Definition A polygon is a closed geometic figue, wit at least tee sides, in wic eac side is a staigt line segment. Te most common polygons ae squaes, ectangles, and tiangles. Ojectives A Find te peimete of a polygon. B Find te cicumfeence of a cicle. C Find te aea of a polygon. D Find te aea of a cicle. Examples now playing at MatTV.com/ooks Definition Te peimete of any polygon is te sum of te lengts of te sides, and it is denoted wit te lette P. To find te peimete of a polygon we add all te lengts of te sides togete. Hee ae te most common polygons, along wit te fomula fo te peimete of eac. squae ectangle tiangle w a c s P = 4s l P = 2l + 2w P = a + + c We can justify ou fomulas as follows. If eac side of a squae is s units long, ten te peimete is found y adding all fou sides togete: Peimete P s s s s 4s Likewise, if a ectangle as a lengt of l and a widt of w, ten to find te peimete we add all fou sides togete: Peimete P l l w w 2l 2w EXAMPLE 1 Find te peimete of te given ectangle. 5 yd PRACTICE PROBLEMS 1. Find te peimete of te ectangle in Example 1 if te lengt and widt ae eac inceased y 1 yad. 8 yd SOLUTION Te given ectangle as a widt of 5 yads and a lengt of 8 yads. We can use te fomula fo P 2l 2w to find te peimete. We ave: P 2(8) 2(5) yads Answe yads 5.10 Aea and Peimete 401 cm_c05.indd 401 8/28/09 5:22:16 PM

402 Capte 5 Geomety B Cicumfeence Te cicumfeence of a cicle is te distance aound te outside, just as te peimete of a polygon is te distance aound te outside.

2 402 Capte 5 Geomety B Cicumfeence Te cicumfeence of a cicle is te distance aound te outside, just as te peimete of a polygon is te distance aound te outside. Te cicumfeence of a cicle can e found y measuing its adius o diamete and ten using te appopiate fomula. Te adius of a cicle is te distance fom te cente of te cicle to te cicle itself. Te adius is denoted y te lette. Te diamete of a cicle is te distance fom one side to te ote, toug te cente. Te diamete is denoted y te lette d. In te figue elow we can see tat te diamete is twice te adius, o d 2 Te elationsip etween te cicumfeence and te diamete o adius is not as ovious. As a matte of fact, it takes some faily complicated matematics to sow just wat te elationsip etween te cicumfeence and te diamete is. C d C = cicumfeence = adius d = diamete If you took a sting and actually measued te cicumfeence of a cicle y wapping te sting aound te cicle and ten measued te diamete of te same cicle, you would fi nd tat te atio of te cicumfeence to te diamete, C/d, would e appoximately equal to Te actual atio of C to d in any cicle is an iational nume. It can t e witten in decimal fom. We use te symol π (Geek pi) to epesent tis atio. In symols, te elationsip etween te cicumfeence and te diamete in any cicle is C π d Knowing wat we do aout te elationsip etween division and multiplication, we can ewite tis fomula as C πd Tis is te fomula fo te cicumfeence of a cicle. Wen we do te actual calculations, we will use te appoximation 3.14 fo π. Because d 2, te same fomula witten in tems of te adius is C 2π 2. Find te cicumfeence fo eac coin in Example 2 fom te dimensions given elow. Round answes to te neaest undedt. a. Diamete 0.92 inces. Radius millimetes EXAMPLE 2 Find te cicumfeence of eac coin. a. 1 Euo coin (Round to te neaest wole nume.) Diamete millimetes cm_c05.indd 402

5.10 Aea and Peimete 403. Susan B. Antony dolla (Round to te neaest undedt.) Radius 0.52 inc SOLUTION Applying ou fomulas fo cicumfeence we ave: a. C πd (3.14)(23.25) 73 mm. C 2π 2(3.14)(0.52) 3.

3 5.10 Aea and Peimete 403. Susan B. Antony dolla (Round to te neaest undedt.) Radius 0.52 inc SOLUTION Applying ou fomulas fo cicumfeence we ave: a. C πd (3.14)(23.25) 73 mm. C 2π 2(3.14)(0.52) 3.27 in. Cicles and te Eat Tee ae many cicles modeled on te suface of te eat. Te most familia ae te latitude and longitude lines. Of tese cicles, te ones wit te lagest cicumfeence ae called geat cicles. All of te longitude lines ae geat cicles. Of te lattitude lines, only te equato is a geat cicle. Pime Meidian Longitude Equato Latitude EXAMPLE 3 If te cicumfeence of te eat is appoximately 24,900 miles at te equato, wat is te diamete of te eat to te neaest 10 miles? SOLUTION We sustitute 24,900 fo C in te fomula C πd, and ten we solve fo d. 3. Te cicumfeence of te moon at te equato is 6,790 miles. To te neaest 10 miles, find te diamete of te moon 24,900 πd 24, d Sustitute 3.14 fo π d 24, Divide eac side y 3.14 d 7,930 miles Answes 2. a in mm 3. Appoximately 2,160 miles cm_c05.indd 403

4 404 Capte 5 Geomety 1 cm 2 1 cm 2 1 cm 2 1 cm 2 1 cm 2 1 cm 2 1 cm 2 1 cm 2 4 cm 2 cm C Aea of a Polygon Recall tat te aea of a fl at oject is a measue of te amount of suface te oject as. Te aea of te ectangle in te magin is 8 squae centimetes, ecause it takes 8 squae centimetes to cove it. As we ave noted peviously, te aea of tis ectangle can also e found y multiplying te lengt and te widt. Aea (lengt) (widt) (4 centimetes) (2 centimetes) (4 2) (centimetes centimetes) 8 squae centimetes Fom tis example, and otes, we conclude tat te aea of any ectangle is te poduct of te lengt and widt. Hee ae te most common geometic fi gues along wit te fomula fo te aea of eac one. Te only fomulas tat ae new to us ae te ones tat accompany te paallelogam and te cicle. Squae Rectangle Paallelogam s w s Aea = (side)(side) = (side) 2 = s 2 l Aea = (lengt)(widt) = lw Aea = (ase)(eigt) = Tiangle Cicle 1 Aea = (ase)(eigt) 2 1 A = 2 Aea = π(adius) 2 A = π 2 4. Find te aea. 3 cm }2 cm EXAMPLE 4 Te paallelogam elow as a ase of 5 centimetes and a eigt of 2 centimetes. Find te aea. 5 cm } 2 cm SOLUTION If we apply ou fomula we ave Aea (ase)(eigt) A cm 2 cm_c05.indd 404

5 5.10 Aea and Peimete 405 O, we could simply count te nume of squae centimetes it takes to cove te oject. Tee ae 8 complete squaes and 4 alf-squaes, giving a total of 10 squaes fo an aea of 10 squae centimetes. Counting te squaes in tis manne elps us see wy te fomula fo te aea of a paallelogam is te poduct of te ase and te eigt. To justify ou fomula in geneal, we simply eaange te pats to fom a ectangle. Paallelogam Rectangle Move tiangle to igt side EXAMPLE 5 Te tiangle elow as a ase of 6 centimetes and a eigt of 3 centimetes. Find te aea. 5. Find te aea. 3 cm }2 cm 4 cm 6 cm SOLUTION If we apply ou fomula we ave Aea 1 2 (ase)(eigt) A cm 2 As was te case in Example 4, we can also count te nume of squae centimetes it takes to cove te tiangle. Tee ae 6 complete squaes and 6 alfsquaes, giving a total of 9 squaes fo an aea of 9 squae centimetes. D Aea of a Cicle EXAMPLE 6 A cicle as a adius of 14.5 millimetes. Find te aea of te cicle to te neaest wole nume. 6. A cicle as a adius of 5 feet. Find te aea. SOLUTION Using ou fomula fo te aea of a cicle, and using 3.14 fo π, we ave A π 2 Fomula fo aea 3.14(14.5) 2 Sustitute in values 3.14(210.25) Squae mm 2 Multiply 660 mm 2 Round to te neaest wole nume Answes 4. 6 cm cm ft 2 cm_c05.indd 405

6 406 Capte 5 Geomety 7. Find te aea of te saded potion of tis figue. EXAMPLE 7 Find te aea of te saded potion of tis figue. 10 cm 12 ft SOLUTION We ave a cicle inscied in a squae. We notice te diamete of te cicle is te same lengt as one side of te squae. To find te aea of te saded egion, we sutact te aea of te cicle fom te aea of te squae as follows: A π π ft 2 Getting Ready fo Class Afte eading toug te peceding section, espond in you own wods and in complete sentences. 1. Wat is te peimete of a polygon? 2. How ae peimete and cicumfeence elated? 3. Wat does π epesent? 4. How do you find te aea of a cicle? Answe cm 2 to te neaest tent cm_c05.indd 406 8/28/09 5:22:18 PM

Essential Question How can you find the surface area and the volume of a cone? 3 in. π

11.7 Suface Aeas and Volumes of Cones Essential Question How can you find te suface aea and te volume of a cone? Finding te Suface Aea of a Cone Wok wit a patne. Constuct a cicle wit a adius of 3 inces.