Total surface area: the area of the lateral faces combined with the area of both bases

Transcription

1 Capte 9: Measuement and te Metic System Section 9.: Volume and Suface Aea Total Suface Aea of a Cylinde and a Pism Lateal aea: te aea of te egions bounded by te lateal faces of a pism Total suface aea: te aea of te lateal faces combined wit te aea of bot bases Aeas of a suface: te lateal suface aea of a given suface is te sum of te aeas of all of te faces o sides (ote tan te bases) of te figue. Te total suface aea is te lateal suface aea added to te aea of te bases of te space figue. Suface aea of a igt ectangula pism: SA ectangula pism = 2(lw + l + w) Suface aea of igt egula n-gonal pism: SA n-gonal pism = P + ap = ns + ans = ns( + a) Suface aea of igt cylinde: SA cylinde = 2π 2 + 2π = 2π( + ) 2(pi)

2 Volume Measue Volume: associates a unique numbe wit eac closed space egion; to find te volume of a pism o cylinde we multiply base aea times eigt; volume is given in cubic units o units cubed 1. Te numbe of squaes in te base aea matces te numbe of cubes o cubic units in te bottom laye 2. Te numbe of segments in te eigt matces te numbe of layes Volume of a igt ectangula pism: Te volume of a igt ectangula pism is te poduct of te pism s lengt, widt, and eigt. You can also tink of it as te aea of te base (Aectangle = lw) times te eigt,, ten we ave V ectangula pism = lw l w Popeties of Volume: 1. Volume is additive; tat is, te volume of te wole is equal to te sum of te volumes of te nonovelapping pats. If R and S ae nonovelapping space egions (possibly wit sufaces in common), ten te volume of R S is equal to te volume of R plus te volume of S. 2. If space egion R as te same size and te same sape as space egion S, ten te volume of space egion R equals te volume of space egion S.. If a space egion is cut into pats and eassembled witout ovelapping to fom anote space egion, ten te space egions ave te same volume. Units of Measue fo Volume Volume: cubic linea measues o 1728 in = 1 ft o Wee do we talk about cubic feet of a substance in te eal wold? swimming pool wate o 27 ft = 1 yd o Wee do we talk about cubic yads of a substance in te eal wold? concete

3 Volume of a Cube: Volume of a cube is found by: V cube = s wee s is te lengt of an edge of te cube o one of te sides of te squae faces Cubic centimete: Sometimes efeed to in common language as a cc; 1 cm = 1 ml wee L is te metic unit fo volume known as lite. Cavaliei s Pinciple Cavaliei s Pinciple: Given: a plane and any two solids wit bases in te plane. If evey plane paallel to te given plane intesects te two solids in coss-sections tat ave te same aea, ten te solids ave te same volume. Volume of a Pism and a Cylinde Oblique Pism: an oblique pism is any pism tat is not a igt pism. Oblique Cylinde: an oblique cylinde is any cylinde tat is not a igt cylinde. Cavaliei s pinciple applied: te fomulas fo finding te volumes of oblique objects emain te same, but te eigt is no longe te lengt of an edge o lateal face. Te eigt is detemined by te pependicula distance between te planes containing te bases. l w Volume of Pism o Cylinde: Te volume of any pism o cylinde is te aea () of a coss-section cut by a plane paallel to te base multiplied by te altitude o eigt (), ten V pism o cylinde = Tiangula Pism: V tiangula pism = = 0.5b b 2

4 Cylinde (tin can): V cylinde = = π 2 Suface Aea and Volume of a Pyamid Regula pyamid: te lateal faces ae isosceles tiangles Slant eigt of pyamid: a pependicula fom te vetex of te pyamid to te base of one of te isosceles tiangles is called te slant eigt (t) of te pyamid. Aea of te lateal faces of a egula pyamid: te aea of eac of te isosceles lateal faces of a egula pyamid is one-alf te slant eigt (t) times an edge (s) of te polygonal base, ten SA one lateal face = 0.5ts; te base as n edges, so te suface aea of all of te lateal faces is SA all lateal faces = 0.5nst = 0.5Pt since P egula n-gon = ns Suface Aea of a Pyamid: fo a egula pyamid wit slant eigt t, peimete of te base P, and aea of te base : SA egula pyamid = 0.5Pt + = 0.5nst + 0.5ans = 0.5ns(t + a) Suface aea squae pyamid: SA = 0.5Pt + = 0.5nst + s 2 = s(0.5nt + s) t s Slant eigt of cone: a pependicula fom te vetex of te cone to te base of te cone is called te slant eigt (t) of te cone. Lateal suface aea of a cone: alf of te poduct of te cicumfeence C and te slant eigt t --- SA lateal cone = 0.5tC = 0.5t(2π) = πt NOTE: numbes ae witten befoe symbols o vaiables, symbols ae witten befoe vaiables, vaiables ae witten in alpabetical ode by convention.

5 Suface Aea Cone: sum of te lateal suface aea and te aea of te base : SA cone = πt + π 2 = π(t + ) t Volume of a Pyamid: V pyamid = s Volume of a Cone: V cone = = π2 Volume and Suface Aea of a Spee Suface Aea of a Spee: SA spee = 4π 2 Volume of a Spee: V spee = 4π Execise Sets: Homewok: p. 46: 1ac, 2,, 5, 7, 9, 11, 12, 1, 14ac, 16, 17, 18, 19, 20ac lazeview: p. 46: 15