1. Sec 4.9 – Circles & Volume Volume of Pyramids & Cones Name: Using Cavalieri’s Principle we can show that the volume of a pyramid is exactly ⅓ the volume of a prism with the same Base and height. Consider a square based pyramid inscribed in cube. Next, translate the peak of the pyramid. Cavalieri’s Principle would suggest that the volume of the oblique pyramid is the same as the original pyramid. Next, we can create 2 more oblique pyramids with the same volume of the original with the remaining space in the cube. In this diagram, we can see the 3 oblique pyramids of equal volume pulled out from the cube. So, this demonstrates a pyramid inscribed in a cube has exactly ⅓ the volume the cube. This idea can be extended to any pyramid or cone. M. Winking Unit 4‐9 page 122

1. Sec 4.9 – Circles & Volume Volume of Pyramids & Cones ... · pyramids with the same volume of the original with the remaining space in the cube. In this diagram, we can see the

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Page 1: 1. Sec 4.9 – Circles & Volume Volume of Pyramids & Cones ... · pyramids with the same volume of the original with the remaining space in the cube. In this diagram, we can see the

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.

M.Winking Unit4‐9page122

1. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).

Volume:

Volume: Volume:

M.Winking Unit4‐9page123

2. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).

Volume:

Volume: Volume:

Findthevolumeoftheregularoctahedron. Findthevolumeoftheirregularsolid.Thebasehasanareaof80cm2andaheightof9cm.

M.Winking Unit4‐9page124

3. FindtheVolumeofthefollowingsolids(figuresmaynotbedrawntoscale).

Volume:

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

Volume:

M.Winking Unit4‐9page125

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= ∙

M.Winking Unit4‐9page126