Surface Area of Revolution

This calculator finds the area swept out when a straight segment rotates about an axis — the lateral surface of a cylinder, cone or conical frustum.

How the calculation works

Pappus's first centroid theorem says a surface of revolution has area S = 2π · ȳ · L: the length of the generating curve times the distance its centroid travels. For a segment running from radius r₁ to radius r₂ the centroid sits at the mean radius (r₁ + r₂)/2, and the length is the slant height l = √(h² + (r₂ − r₁)²). That gives S = π(r₁ + r₂) · l, the same answer as the calculus formula S = 2π∫ y√(1 + y′²) dx.

r₁ and r₂ are the distances from the axis to the two ends of the segment and h is its length along the axis, all in metres; the area is in m². With r₁ = r₂ you get a cylinder, with r₁ = 0 a cone and with h = 0 a flat ring. The tool also reports the slant height, the area including both circular ends, π(r₁² + r₂²), and the solid's volume πh(r₁² + r₁r₂ + r₂²)/3.

Worked example

The defaults r₁ = r₂ = 1 m and h = 2 m describe a cylinder. Slant height l = 2 m, lateral area S = π × (1 + 1) × 2 ≈ 12.566 m². Adding the two ends of π m² each gives ≈ 18.850 m², and the volume is ≈ 6.283 m³. For a cone with r₁ = 0, r₂ = 3 m and h = 4 m the slant height is 5 m and S = π × 3 × 5 ≈ 47.12 m².

Things to keep in mind

• Only straight segments are handled. For a curved profile such as a vase, split it into short segments, compute each one and add them up; finer splitting means better accuracy.
• Spheres and tori have exact formulas — 4πR² for a sphere, 4π²Rr for a torus — which beat a segment approximation.
• When estimating paint or sheet material, use the total-area figure if the ends are covered too, and add an allowance for seams and overlaps.
• Keep all dimensions in one unit: enter centimetres and the area comes out in cm², the volume in cm³.