don steward
mathematics teaching 10 ~ 16

Thursday, 25 July 2013

cone, sphere, cylinder

the solids have a common largest circle radius and a common height

what is the ratio of volumes of
cone : sphere : cylinder
in its simplest form?

establish that the volume of the sphere plus the cone make the volume of the cylinder

(image on Wikimedia)

this result was established by Archimedes using small slices, as explained in 'Cut the Knot' and illustrated here

it can be shown that the volume of the thin disk is the same as the volume of the ring

there's an interesting illustration below (also the above diagram) and an explanation of Archimedes' method on a blog by Zachary Abel

cone  +  hemisphere = cylinder

which solid has the smallest volume?

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