Pappus's Centroid Theorem — Definition, Formula & Examples

Pappus's Centroid Theorem gives you the surface area or volume of a solid of revolution by multiplying the length (or area) of the generating shape by the distance its centroid travels during rotation.

The theorem has two parts. First, the surface area of a surface of revolution generated by rotating a plane curve about an external axis equals $2\pi \bar{y} L$ , where $\bar{y}$ is the distance from the curve's centroid to the axis and $L$ is the arc length. Second, the volume of a solid of revolution generated by rotating a plane region about an external axis equals $2\pi \bar{y} A$ , where $\bar{y}$ is the distance from the region's centroid to the axis and $A$ is the area of the region.

Key Formula

V = 2\pi \bar{y}\, A \qquad \text{and} \qquad S = 2\pi \bar{y}\, L

Where:

$V$ = Volume of the solid of revolution
$S$ = Surface area of the surface of revolution
$\bar{y}$ = Distance from the centroid of the region (or curve) to the axis of rotation
$A$ = Area of the plane region being rotated
$L$ = Arc length of the plane curve being rotated

How It Works

To use the theorem, identify the shape being rotated and the axis of rotation. Find the centroid of the shape (its geometric center) and measure how far that centroid is from the axis. For volume, multiply the region's area by

2\pi

times the centroid distance. For surface area, multiply the curve's arc length by

2\pi

times the centroid distance. The key insight is that

2\pi \bar{y}

equals the total distance the centroid travels in one full revolution around the axis.

Worked Example

Problem: Find the volume of a torus (donut shape) formed by rotating a circle of radius 3 about an axis 5 units from the circle's center.

Step 1: Identify the generating region and its centroid distance. The region is a circle of radius 3, so its area is

\pi r^2

. The centroid of a circle is its center, which is 5 units from the axis.

A = \pi(3)^2 = 9\pi, \qquad \bar{y} = 5

Step 2: Apply Pappus's volume formula.

V = 2\pi \bar{y}\, A = 2\pi(5)(9\pi) = 90\pi^2

Answer: The volume of the torus is

90\pi^2 \approx 888.3

cubic units.

Why It Matters

Pappus's theorem lets you compute volumes and surface areas of solids of revolution without setting up integrals, which is especially useful in engineering design and multivariable calculus. It also works in reverse: if you know the volume of a solid of revolution, you can solve for the centroid of the generating region.

Common Mistakes

Mistake: Using the centroid of the curve for the volume formula (or the centroid of the region for the surface area formula).

Correction: Volume uses the centroid of the filled region and its area

A

. Surface area uses the centroid of the boundary curve and its arc length

L

. These centroids are generally at different distances from the axis.