Where Does The 1/3 come from in the formula for the volume of a cone?

Question

Or what is a web site I can check to find the history of the formula for the volume of a cone?

Answer 1

Actually, geometers determined the formula for the volume of a cone centuries before the calculus was invented. In fact, Archimedes made use of the formula for the volume of a cone when he discovered the formula for determining the volume of a sphere. He did this through rigorous analysis and comparison of the areas of cross sections of a plane which cut across a sphere of a given diameter, d, and a cylinder with the same diameter and height equal to the diameter. Archimedes methods were the ancient precursors to the development of the calculus.

I imagine that the first determination of the volume of a cone came through simple experimentaton and observation. All that the ancients had to do was make a cylinder and a cone shaped container of the same circular base diameter and the same height and then use the cone shaped container to fill the cylinder. The simple observation that it took three containers full of liquid from the cone shape to fill the cylinder was all they needed to deduce the formula. Since the formula for the volume of a cylinder was known to be V = π r² h and it took 3 cones of liquid to fill the cylinder, then all they had to do was set up this ratio to solve for the formula for the volume of the cone:

V (cone) / V (cyl) = x / π r² h.

Realizing that V (cyl) = 3 V (cone) from their observations, they could make that substitution in their ratio to obtain this:

V (cone) / 3 V (cone) = x / π r² h
1 / 3 = x / π r² h

Then all they had to do was solve for x to find their formula:

Cross-multiplying:

3x = π r² h
x = (1/3) π r² h.

That's all there is to it empirically. Proving it mathematically however is not so easy a task.

To find out more about the volume of a cone, just google it and variations thereof.

Answer 2

I think it has to do with that a cone with the same dimensions of a cylinder is exactly one/third of that area. But I might be wrong.