Volume of Pyramid with Square Base

Calculus 1 · Axiom Academy

EXAMPLE Volume of Pyramid with Square Base Derive the classic pyramid volume formula by integrating square cross-sectional areas A solid pyramid has a square base of side b and height h . By slicing it into thin square cross-sections perpendicular to its axis, derive a formula for its volume V in terms of b and h . Stand the pyramid on its apex so the apex is at x=0 and the base at x=h . At distance x from the apex the cross-section is a square; similar triangles fix its side length s(x) . We integrate these square slices from x=0 to x=h . Nice work — you derived the pyramid volume formula straight from calculus, no formula memorized. Cross-section method: , where A(x) is the area of the slice at x . Similar triangles: they pin down how a slice's size scales with x — here , so . The result: , with base area B = b^2 — the same that works for any pyramid or cone. Whenever you can describe a solid by the area of its slices, this same integral gives the volume.

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