Pappus Centroid Theorems

Visualize how to calculate the volume and surface area of solids of revolution using Pappus's Centroid Theorems.

Pappus Centroid Theorems

Concept Overview

Pappus's Centroid Theorems are results from geometry that relate the surface area and volume of solids of revolution to the centroid of the generating region. They provide an alternative to direct integration. For the disk and rectangle used here, symmetry places both the area centroid and the boundary centroid at (0, R), so the same centroid distance applies to both theorems.

First Theorem: Surface Area

The first theorem states that the surface area S of a surface of revolution generated by rotating a curve about an external axis is equal to the product of the curve's arc length C and the distance traveled by its geometric centroid.

S = 2πR × C

Where R is the distance from the axis of revolution to the centroid of the curve, and 2πR is the distance traveled by the centroid.

Second Theorem: Volume

The second theorem states that the volume V of a solid of revolution generated by rotating a planar region about an external axis is equal to the product of the region's area A and the distance traveled by its geometric centroid.

V = 2πR × A

Where R is the distance from the axis of revolution to the centroid of the region, and 2πR is the distance traveled by the centroid.

Worked Example

Let's calculate the volume and surface area of a torus generated by rotating a disk.

Disk radius r = 1
Distance from axis to centroid R = 2

First, we find the area and perimeter of the generating disk:

Area A = πr² = π(1)² = π ≈ 3.141593
Perimeter C = 2πr = 2π(1) = 2π ≈ 6.283185

The distance traveled by the centroid is:

Distance = 2πR = 2π(2) = 4π ≈ 12.56637

Now, we apply the theorems to find the volume and surface area:

Volume V = Distance × A = 4π × π = 4π² ≈ 39.47842
Surface Area S = Distance × C = 4π × 2π = 8π² ≈ 78.95684

Experiment Steps

Set Generating shape to Disk, Half-height or radius r to 1 and Centroid distance R to 2. Observe volume 39.47842 and surface area 78.95684.
Read the dashed centroid orbit in the oblique diagram. It is a circle in the yz-plane about the x-axis, shown as an ellipse by projection, and its length is 4π.
Switch Generating shape to Rectangle and set Rectangle width w to 3. Keeping r = 1 and R = 2, check area 6, perimeter 10, volume 75.39822 and full surface area 125.6637.
Set Centroid distance R to 1. The tangent-axis boundary R = r is rejected in this experiment; no dependent results remain. Restore R to 2.
Drag the diagram to pan, then press Reset view to restore its framing. Length controls retain their values.

Link to module: Pappus Centroid Theorems

Limitations

This experiment requires R > r and positive dimensions, so the axis is strictly external. It deliberately excludes the tangent case R = r; that domain rule is not a claim that every boundary version of the theorem fails there. If an axis crosses the generating region, a swept point can be counted more than once, so the displayed formulas cannot be applied blindly. The diagram is an oblique projection, not a change of the centroid orbit into an ellipse in space. Displayed results use seven significant digits; calculations retain unrounded values. Consistent length units give area and surface in squared units and volume in cubed units.

References

Explore these sources to learn more about the mathematics behind these theorems:

OpenStax Calculus Volume 1: Moments and Centers of Mass explains centroid symmetry and the volume theorem.
Pappus's Centroid Theorems (University of Florida) derives both centroid relations and applies them to a torus.

Pappus Centroid Theorems