This is sad,

The way I remember it is that the volume of a pyramid is 1/3 of the height times the base. And the same rule, incidentally, applies to cones, which makes sense, as it applies to octagonal, hexagonal, and pentagonal pyramids, for example, as well as to square ones.

Still, it's just memorization, which is awkward.

After having learned calculus, though, one gains the ability to derive the formula - at least for the square pyramid.

The derivative of x to the n-th power, with respect to x, is n times x to the (n-1)-th power. A very important formula to remember, because it has lots of uses.

The antiderivative, or the integral, reverses this rule (except for 1/x, whose antiderivative is the natural logarithm of x; this happens because it would be 0/0 otherwise).

So if something falls at an acceleration of 32 feet per second squared, not only do we know the obvious fact that its velocity at time t from being let go will be t times 32 feet per second, but we also know that its position will be t squared times 16 feet. (Derivative of 16 t squared is 2 times 16 t.)

Apply that to a square pyramid whose height is the same as the sides of its base. So the derivative of its volume, as we go down from the tip, is the area of its base, h squared. As the derivative of h cubed is 3 times h squared, the integral of h squared is 1/3 h cubed.