1. Sec 4.9 – Circles & Volume Volume of Pyramids & Cones  Name:         

UsingCavalieri’sPrinciplewecanshowthatthevolumeofapyramidisexactly⅓thevolumeofaprismwiththesameBaseandheight.

Considerasquarebasedpyramidinscribedincube.

Next, translatethepeakofthepyramid.Cavalieri’sPrinciplewouldsuggestthatthevolumeoftheobliquepyramidisthesameas

theoriginalpyramid.

Next,wecancreate2moreobliquepyramidswiththesamevolumeoftheoriginalwiththeremaining

spaceinthecube.

Inthisdiagram,wecanseethe3obliquepyramidsofequalvolumepulledoutfromthecube.So,thisdemonstratesapyramidinscribedinacubehasexactly⅓thevolumethecube.

Thisideacanbeextendedtoanypyramidorcone.

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1. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).

Volume:

Volume:

Volume: Volume:

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2. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).

Volume:

Volume:

Volume: Volume:

Findthevolumeoftheregularoctahedron. Findthevolumeoftheirregularsolid.Thebasehasanareaof80cm2andaheightof9cm.

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3. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).

Volume:

Volume:

ConsidertriangleABCwithverticesatA(0,0),B(4,6),andC(0,6)plottedandacoordinategrid.Determinethevolumeofthesolidcreatedbyrotatingthetrianglearoundthey‐axis.

Volume:

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UsingCavalieri’sPrinciplewecanshowthatthevolumeofaspherecanbefoundby ∙

First,considerahemispherewitharadiusofR.CreateacylinderthathasabasewiththesameradiusRandaheightequaltotheradiusR.Then,removeaconefromthecylinderthathasthesamebaseandheight.

Next,consideracrosssectionthatisparalleltothebaseandcutsthroughbothsolidsusingthesameplane.

Cavalieri’sPrinciplesuggestsifthe2crosssectionshavethesameareathenthe2solidsmusthavethesamevolume.

Theareaofthecrosssectionofthesphereis:∙

UsingthePythagoreantheoremweknow: or

So,withsimplesubstitution:∙

Theareaofthecrosssectionofthesecondsolid is:∙ ∙

Usingsimilartrianglesweknowthath=bandthen,usingsimplesubstitution

∙ ∙

VolumeofHemisphere=VolumeofCylinder–VolumeofCone= ∙ ∙

WealsoknowthatR=b=h.So,VolumeofHemisphere= ∙ ∙ ∙ ∙ ∙

Tofindthevolumeofacompletesphere,wecanjustdoublethehemisphere:VolumeofSphere= ∙

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