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Cylinder Volume Derivation

Where does V = πr²h come from? This page walks through the mathematical derivation step by step — from the area of a circle to stacking cross-sections, using Cavalieri's principle and basic integration. Understand the formula, don't just memorize it.

Volume Derivation Calculator

V = πr²h
Stack circular slices Each slice: πr² × Δh → V = πr²h

From Circle Area to Cylinder Volume

Circle → Cylinder

The area of a circle is A = πr². This comes from dividing the circle into infinitely many thin triangular wedges and rearranging them into a rectangle with width πr and height r, giving area πr².

A cylinder is formed by stacking identical circles along the height axis. If every horizontal cross-section from bottom (y = 0) to top (y = h) is the same circle with area πr², the total volume is the sum of all those slices:

V = πr² × h

This is the same logic as finding the volume of any prism: base area × height.

Cavalieri's Principle

A ⟷ B Cavalieri's Principle

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.

Proof by Integration

∫₀ʰ πr² dy = πr²h

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.

Frequently Asked Questions

Why is cylinder volume πr²h?
The base is a circle with area πr². The cylinder stacks that circle uniformly through height h. Volume = base area × height = πr²h.
What is Cavalieri's principle?
If two solids have the same cross-sectional area at every height, they have the same volume. It lets us compare volumes without integration.
Does the derivation work for oblique cylinders?
Yes. By Cavalieri's principle, an oblique cylinder has the same cross-sections as a right cylinder, so the volume is the same: V = πr²h (with h as the perpendicular height).
How is the area of a circle πr² derived?
Cut the circle into thin ring strips (annuli) from center to edge. Each ring at distance x from center has circumference 2πx and width dx, giving area 2πx·dx. Integrate from 0 to r: ∫₀ʳ 2πx dx = πr².
What is the disk method in calculus?
The disk method finds the volume of a solid of revolution by integrating the area of circular cross-sections: V = ∫ π[r(x)]² dx. For a cylinder, r(x) is constant, giving V = πr²h.