May 06 2017, 19:15

I’m going to ask a ridiculously embarrassing question. I’ve forgotten math. Why isn’t the volume of a cone equal to the area of a triangle times the circumference of the base? Circle area — radius times circumference? I know it’s not so, but can’t explain to my daughter why:(

Update: Explained it like this:

You can’t spin a line segment 2π times because the density of points near the center and at the edge of the circle will be different. It’s correct to find the area of a circle by adding the lengths of concentric circles from 0 to R. If you represent this as a graph, you get a triangle with an area of 2πR*R. The same approach applies to the cone, but you need to sum the areas of cone slices from a base area of zero.

But unlike the circle area example, here the relationship isn’t linear, it’s parabolic, and you can’t find the area of a parabolic segment “on your fingers”; you need to explain what an integral is.

We relate the cone’s height and slice radius to a single variable of the integral, and find it, which gives us the volume of the cone. The integral is taken from the second power, hence the three in the denominator after calculating the integral. Explaining “on the fingers” why integrals are computed from functions x^n is a topic of its own, it’s better to just believe it for now)