Geometry Supplement for Math 60 Perimeter, Area, and Volume

Transcription

1 Geomety Suppement fo Mat 60 Peimete, Aea, and Voume Geomety comes fom te Geek wods geo meaning eat and meton meaning measue. Today we wi ean about ow to measue cetain featues of two- and tee-dimensiona figues tat we see commony in ou eveyday wod. Peimete Te peimete of a two-dimensiona figue is te measue of te distance aound tat figue. Peimete is measued in inea units, suc as feet o metes. Te peimete of a staigt-ine figue can be cacuated by adding te engts of te sides of te figue. In te exampe beow, we see tat te peimete is 16.5 metes by totaing te engts of a five sides. Exampe 1: Find te peimete of te figue. Peimete of te figue P = 4 m +.5 m + m + 5 m + m P = 16.5 m Te peimete of te figue is 16.5 metes. Rectanges ae used fequenty enoug tat we usuay efe to a fomua instead of te definition wen speaking of tei peimetes. Te fomua fo te peimete of a ectange is P = + w wee is te engt and w is te widt of te ectange. You can easiy see wy beow: Peimete of te ectange P = + w + + w w w P = + w A cice is defined as a set of points in te pane tat ae equay distant fom a given point, te cente. Te adius of a cice is a ine segment fom te cente to any point on te cice. Te diamete of a cice is a ine segment toug te cente in wic its endpoints ie on te cice. d Note: d = Cicumfeence of a cice C = π d o C = π Te peimete of a cice is caed its cicumfeence. Te cicumfeence of a cice is just ove tee times te engt of te diamete. Moe pecisey, te cicumfeence is te poduct of π and te diamete engt, C = π d. It can aso be cacuated using te adius: C = π. You can ceck tis out yousef wit a can fom you kitcen cupboad and a piece of sting. Just cut te sting so tat its engt is one time aound te can (cicumfeence). Ten count off te diamete engts you ave witin te piece of sting by oding it acoss te top of te can. Regadess of te size of te can, you wi aways be abe to count off tee diametes wit a itte bit eft ove.

2 Exampe : Find te cicumfeence of te cice. Expess it in tems of π. Ten ound to te neaest tent. 5 in C = π C = π (5 inces) C = 10π inces C! 1.4 inces Te cicumfeence is exacty 10π inces and appoximatey 1.4 inces. Exampe : Find te peimete of te figue beow in wic a ectange and a semicice ae joined. (Round to te neaest tent.) Peimete = tee sides of te ectange + te cicumfeence of te semicice Peimete = (1 ft + 8 ft + 1 ft) + ½ [π(8 ft)] Peimete = ft + 4π ft Peimete! 44.6 feet Te peimete of te figue is appoximatey 44.6 feet. Exampe 4: (a) Find te peimete of a ectange tat as a widt of 4 yads and a engt of 9 yads. (b) Find te cicumfeence of a cice tat as a diamete of 1 centimetes. Exampe 5: How many pants spaced evey 6 inces ae needed to suound a cicua gaden wit a 0-foot adius?

3 Exampe 6: Te peimete of a yied sign in te sape of an isoscees tiange is feet. If te sotest side is feet ess tan te ote two sides, find te engt of te sotest side. YIELD Aea Te aea of a two-dimensiona figue is te measue of te space inside tat figue. We measue te space inside a figue using unit squaes. Reca tat a squae as te same engt and widt. Te engt and widt of a unit squae ae eac one. Fo instance, ow many squae inces (eac squae as sides equa to 1 inc) can fit into a ectange tat as a engt of 7 inces and a widt of 4 inces as seen in te figue beow? A gid sowing squae inces as been oveaid on te ectange so tat we can count te numbe of squaes contained witin it. You may simpy count tem and find tat te aea is 8 squae inces. Anote way to count te squaes is to notice tat tee ae seven coumns of fou squaes eac. Tis is a cassic mutipication pobem. Seven goups of fou can be stated as 7 times 4 yieding an aea of 8 squae inces. Fom tis simpe exampe we can see tat te aea of a ectange wi be te poduct of its engt and widt. We can aso find aea fomuas fo ote basic twodimensiona figues: Rectange Tiange Cice Tapezoid 1 A = w A = b A =! A = 1 ( a + b ) a w b b

4 Exampe 7: (a) Find te aea of te foowing figues using te fomua given fo its aea. (b) Confim te aea by appoximating te numbe of squaes inside te figue. (Matc up patia squaes to appoximate woe squaes.) (a) Tiange: 1 A = b A = 1 (6 m)(4 m ) A = 1 m Te aea is 1 squae metes. (a) Cice: A =! A =! ( m) A = 9" m! 8. m Te aea is! 8. squae metes. (b) Tee ae 6 woe squaes and (b) Tee ae 16 woe squae and 0 1 patia squaes. Matcing up patia squaes. Matcing up patia patia squaes, we get 6 woes. squaes, we get about 1 woe squaes. Teefoe, te aea is made up So te aea is made up of about 8 of 1 woe squaes. Tis woe squaes. Tis matces up we matces ou aea in pat a. wit te aea found in pat a. Exampe 8: Te widt of a ectangua gaden is 5 feet. Te aea of te gaden is 150 squae feet. Find te engt of te gaden. Exampe 9: Te cicumfeence of a cice is 14π inces. Find te cice s diamete, adius, and aea.

5 Exampe 10: Hadwood fooing costs $10 pe squae foot. How muc wi it cost (to te neaest doa) to cove te dance foo sown in te figue wit adwood fooing? 0 ft 60 ft Exampe 11: Find te aea of te figue beow.

6 Voume A compute box, a basketba, and soup can ae exampes of tee-dimensiona figues. Voume is te measue of te amount of space inside suc an object. Voume is measued using unit cubes. Reca tat a cube as te same engt, widt, and eigt. Suga cubes used in tea o coffee and apabet bocks tat kids (and aduts ) pay wit ae exampes of cubes. Fo instance, conside a box tat as a widt of 5 inces, a engt of 9 inces, and a eigt of inces. Imagine ceating te same size box wit inc cubes. How many cubes woud make 9 in in 5 in Voume of te Box V = w V = ( 9 in)(5 in)( in) V = 15 in up te box? Te box is inces ig so tee wi be ayes of cubes witin te box. Eac aye contains 5 ows of 9 cubes. So tat makes 45 cubes pe aye. Since tee ae ayes of 45 cubes eac inside te box, te box contains 15 cubes. Tus we concude tat te voume of te soe box is 15 cubic inces. Note tat we mutipied te engt, widt, and eigt togete to get te voume of te box. Te fomua fo te voume of a ectangua soid is V = w. Hee ae some ote common voume fomuas: Rectangua Soid Cyinde Spee Cone V = w V =! 4 V =! V = 1! w

7 Exampe 1: Find te voume of soup can tat as a eigt of 1 centimetes and a adius of 4 centimete. 4 cm SOUP 1 cm Voume of te Soup Can V =! V =! (4 cm) (1 cm) V =! (16 cm )(1 cm) V = 19" cm! 60. cm Te voume of te soup can is 19π cubic centimetes, appoximatey 60. cubic centimetes. Exampe 1: A wate stoage tank is in te sape of a cone wit te pointed end down. If te adius is 14 feet and te dept of te tank is 15 feet, appoximate te voume of te tank (ound to te neaest tent). Exampe 14: A cyindica fis tank as a diamete of 6 feet and a eigt of feet. How many topica fis can be put in te tank if eac fis needs 5 cubic feet of wate? Exampe 15: Andea as dug a ectangua foundation metes by metes fo a new sed in e backyad. Se as emoved 9 cubic metes of dit. How deep is te oe?