Volume harmony

Take a cone, sphere, and cylinder of equal height and radius. The volume of the cone plus the volume of the sphere is equal to the volume of the cylinder.

Volumes
Cmglee / CC BY-SA

I don’t have anything clever to say about this harmony, it just really annoys me that sometimes math is so neat and clean.

  • The volume of a sphere is 4/3 times πr3, where r is the radius.
  • The volume of a cone is 1/3 times πr2h, where h is the height.
  • The volume of a cylinder is πr2h.

And here’s the trick: the height of the sphere is the same as its diameter, so it’s twice the radius. If you take a sphere and a cone and a cylinder of equal radius and equal height, all of their heights are twice their radii. Swap h (height) in those calculations above for 2r, and you get the following:

  • The sphere’s volume remains at 4/3 πr2.
  • The cone’s volume is 1/3 πr22r… simplified to 2/3 πr3.
  • The cylinder’s volume is πr22r… simplified to 2 πr3. A little fancy math footwork, and you get this: 6/3 πr3

πr3 shows up in all three calculations, and if we look at the numbers before it there’s a clear and direct ratio between the three:

Cylinder : sphere : cone
6/3 : 4/3 : 2/3

Or, to put it more plainly:

Cylinder : sphere : cone
3 : 2 : 1

In other words, the volume of the sphere is twice that of the cone, and the volume of the cylinder is three times that of the cone. Cone plus sphere equals cylinder.

[Thanks to Futility Closet for writing about this one.]

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