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Pappus’s Theorem

Pappus’s Theorem
Theorem of Pappus

A method for finding the volume of a solid of revolution. The volume equals the product of the area of the region being rotated times the distance traveled by the centroid of the region in one rotation.

Key Formula

V=2πrˉ AV = 2\pi \bar{r} \, A
Where:
  • VV = Volume of the solid of revolution
  • AA = Area of the plane region being rotated
  • rˉ\bar{r} = Distance from the centroid of the region to the axis of rotation
  • 2πrˉ2\pi \bar{r} = Distance the centroid travels in one full rotation (circumference of its circular path)

Worked Example

Problem: Find the volume of the torus (donut shape) formed by rotating a circle of radius 3 about an axis that is 5 units from the center of the circle.
Step 1: Identify the area of the region being rotated. The region is a circle of radius 3.
A=πr2=π(3)2=9πA = \pi r^2 = \pi(3)^2 = 9\pi
Step 2: Identify the distance from the centroid of the region to the axis of rotation. The centroid of a circle is its center, which is 5 units from the axis.
rˉ=5\bar{r} = 5
Step 3: Apply Pappus's Theorem. Multiply the area by the distance the centroid travels in one full rotation.
V=2πrˉ A=2π(5)(9π)=90π2V = 2\pi \bar{r} \, A = 2\pi(5)(9\pi) = 90\pi^2
Answer: The volume of the torus is 90π2≈888.390\pi^2 \approx 888.3 cubic units.

Another Example

Problem: A semicircular region of radius 4 has its diameter along the y-axis. The region is rotated about the y-axis. Use Pappus's Theorem to find the volume of the resulting solid (a sphere).
Step 1: Find the area of the semicircular region.
A=12πr2=12π(4)2=8πA = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi(4)^2 = 8\pi
Step 2: The centroid of a semicircle of radius r lies at a distance of 4r/(3π) from the diameter. Here r = 4.
rˉ=4(4)3π=163π\bar{r} = \frac{4(4)}{3\pi} = \frac{16}{3\pi}
Step 3: Apply Pappus's Theorem.
V=2πrˉ A=2π⋅163π⋅8π=256π3V = 2\pi \bar{r} \, A = 2\pi \cdot \frac{16}{3\pi} \cdot 8\pi = \frac{256\pi}{3}
Step 4: Verify: this matches the standard sphere volume formula with radius 4.
V=43π(4)3=256π3✓V = \frac{4}{3}\pi(4)^3 = \frac{256\pi}{3} \checkmark
Answer: The volume is 256π3≈268.1\frac{256\pi}{3} \approx 268.1 cubic units, confirming the sphere volume formula.

Frequently Asked Questions

Does Pappus's Theorem also work for surface area?
Yes. There is a second part of Pappus's Theorem for surface area: the surface area of a solid of revolution equals the length of the curve being rotated multiplied by the distance its centroid travels, giving S=2πrˉ LS = 2\pi \bar{r} \, L, where LL is the arc length. The version for volume uses area; the version for surface area uses arc length.
Can you use Pappus's Theorem to find the centroid if you already know the volume?
Absolutely. If you know both the volume and the area of the region, you can solve for the centroid distance: rˉ=V/(2πA)\bar{r} = V / (2\pi A). This reverse application is a common technique for locating centroids of regions whose volumes of revolution are known.

Pappus's Theorem vs. Disk/Washer Method

The disk/washer method requires setting up and evaluating an integral. Pappus's Theorem bypasses integration entirely — you only need the area of the region and the location of its centroid. Pappus's Theorem is especially powerful when the centroid is known (or easy to find) but the integral would be complicated.

Why It Matters

Pappus's Theorem turns difficult volume integrals into simple multiplication once you know the centroid. It is widely used in engineering and physics to compute volumes and surface areas of symmetric objects like pipes, gaskets, and tori. It also works in reverse — if you know the volume, you can determine where the centroid is, which is valuable in structural analysis.

Common Mistakes

Mistake: Using the distance from the axis to the edge of the region instead of the distance to the centroid.
Correction: The theorem specifically requires the distance from the axis of rotation to the centroid of the region, not to any other point. For a rectangle, this is the center; for a triangle, it is one-third of the way from each side.
Mistake: Forgetting the factor of 2π (writing V = r̄A instead of V = 2πr̄A).
Correction: The centroid distance r̄ is a radius, not a circumference. You must multiply by 2π to convert it into the distance the centroid actually travels around the axis.

Related Terms

  • Volume — The quantity Pappus's Theorem computes
  • Solid of Revolution — The type of solid this theorem applies to
  • Centroid — Key geometric point used in the formula
  • Disk Method — Alternative integration technique for volumes
  • Washer Method — Alternative method for hollow solids of revolution
  • Shell Method — Another integration alternative for revolution volumes
  • Torus — Classic shape whose volume is found via Pappus