Cylinder Volume Derivation

Cylinder Volume Derivation is an educational resource that explains where the formula V = πr²h comes from — not just what it is, but why it works. This page exists because understanding the derivation gives you deeper mathematical insight, helps you remember the formula, and enables you to adapt it for non-standard shapes.

The derivation traces the formula from first principles: starting with the area of a circle (πr²), then extending it into three dimensions by stacking those circles along the height axis. It uses Cavalieri's principle and basic integration to prove the result rigorously.

This is invaluable for math students studying mensuration and calculus, teachers preparing lessons on solid geometry, and anyone who wants to understand the formula rather than just memorize it.

Cavalieri's principle states: if two solids have equal cross-sectional areas at every height, they have equal volumes. This is why oblique cylinders have the same volume as right cylinders — tilting doesn't change the cross-sections.

Imagine a tall stack of coins. Whether the stack is straight or leaning, the total volume of metal is the same. Each coin (cross-section) has the same area regardless of the tilt.

This principle also explains why V = πr²h works even when the cylinder is slanted, as long as h is the perpendicular height between the bases.

Using calculus, place the cylinder along the y-axis from 0 to h. At each height y, the cross-section is a circle of radius r, with area A(y) = πr².

The volume is the integral: V = ∫₀ʰ A(y) dy = ∫₀ʰ πr² dy = πr² × [y]₀ʰ = πr²h

Since r is constant (it doesn't depend on y), the integral is straightforward. This is one of the simplest applications of the disk method in calculus.