The Volume of a Cone
As every reader of this blog undoubtedly knows, the volume of a cone with height H and radius R is 1/3*pi*R^2*H. As a reader of this blog, of course I also knew this - but I found myself unable to prove it. Until a few nights ago. Inspired by the classic book "Calculus Made Easy" by Silvanus P. Thompson and updated by Martin Gardner, which I am currently reading, I attempted to rediscover the simple proof for myself.
I jot it down here so that twenty years from now, I need only to recover this blog entry to remind myself of the proof.
Imagine a triangle with (x, y) coordinates (0,0), (H, 0), and (H, R). Now, imagine this triangle rotating around the X-axis. You will see that this forms a cone with the specified dimensions. Our goal is to determine the volume of this cone. Imagine a little strip of that triangle, with an infinitely small width and height y. When this is rotated around the X-axis, it will form a circle, with the area pi*y^2, as the radius is y. To find the area of the cone, we need to add all these little circles together as x goes from 0 to H - e.g. a little tiny circle at the top of the cone to the full big radius-R circle at the bottom of the cone. (Since my cone is "lying down" it is really from the left to the right - but I think you get the picture).
So, we have to take the integral of 1/2*pi*y^2 from x=0 to x=H. First we must figure out the value of y, in terms of x. At any point x along the x axis, the triangle formed by (0,0), (x, 0), (x, y) must be geometrically similar (e.g. proportional) to the triangle (0, 0), (H, 0), (H, R), as the two triangles have the same three angles, they must be similar. Thus H/R = x/y. Or rewritten as y/x = R/H or y = R*x/H. So, we know have to integrate:
pi*(R*x/H)^2 which is the same as itegrating
pi*R^2/H^2 * the integral of x^2 taken from 0 to H.
The integral of x^2 is x^3/3, taking the value at H, we have H^3/3 and at 0, we have 0. So the integral is H^3/3. Now multiplying by the constants from earlier, we have to solve
pi*R^2/H^2 * H^3/3 which yields
pi*R^2*H/3 or 1/3*pi*R^2*H.
Apologies for horrible mathematical formatting!