What is the relationship of the volume between a sphere and a cylinder of the same dimensions?

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=> The volume of a sphere is 2/3 of the volume of a cylinder with same radius, and height equal to the diameter.

NOTE: The formula for the volume of a sphere is 4⁄3πr³. For a cylinder, the formula is πr²h. A cone is ⅓ the volume of a cylinder, or 1⁄3πr²h.

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Volume of a Cone vs Cylinder

Let's fit a cylinder around a cone.

The volume formulas for cones and cylinders are very similar:

So the cone's volume is exactly one third ( 1 3 ) of a cylinder's volume.

(Try to imagine 3 cones fitting inside a cylinder, if you can!)

Volume of a Sphere vs Cylinder

Now let's fit a cylinder around a sphere .

We must now make the cylinder's height 2r so the sphere fits perfectly inside.

So the sphere's volume is 4 3 vs 2 for the cylinder

Or more simply the sphere's volume is 2 3 of the cylinder's volume!

The Result

And so we get this amazing thing that the volume of a cone and sphere together make a cylinder (assuming they fit each other perfectly, so h=2r):

Isn't mathematics wonderful?

Question: what is the relationship between the volume of a cone and half a sphere (a hemisphere)?

Surface Area

What about their surface areas?

No, it does not work for the cone.

But we do get the same relationship for the sphere and cylinder (2 3 vs 1)

And there is another interesting thing: if we remove the two ends of the cylinder then its surface area is exactly the same as the sphere:

Which means that we could reshape a cylinder (of height 2r and without its ends) to fit perfectly on a sphere (of radius r):

(Research "Archimedes' Hat-Box Theorem" to learn more.)

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