Pappus's Theorem
College/University
Definition
A geometry theorem that helps you identify the surface area and volume of surfaces of a solid of revolution. It uses the distance travelled by the centroids of a curve and the region that experiences revolution. Named after Pappus of Alexandria who is attributed to this theorem.
Worked examples
\(S = 2\pi \bar{y} L\)
Surface area of revolution: arc length L times the distance \(2\pi \bar{y}\) its centroid travels around the axis.
\(V = 2\pi \bar{x} A\)
Volume of revolution: area A times the distance \(2\pi \bar{x}\) its centroid travels around the axis.
Common mistakes
- \(V = 2\pi \bar{x} L\) → \(V = 2\pi \bar{x} A\) Volume uses the area A of the region, not the arc length L of the curve.
- Using the centroid of the solid instead of the centroid of the original region → Use the centroid \((\bar{x}, \bar{y})\) of the flat region before rotation Pappus's Theorem requires the centroid of the region being revolved, not the final 3D solid.
- \(S = \pi \bar{y} L\) → \(S = 2\pi \bar{y} L\) The centroid travels the full circumference \(2\pi \bar{y}\), not half.
Where you'll use it next
You'll apply Pappus's Theorem in multivariable calculus when computing volumes and surface areas of solids of revolution, especially for complex shapes where direct integration is cumbersome.
Found in 1 StudyPug lesson
Mastering the Disc Method for Volumes of Revolution
UniversityUniversityCalculus 2
Unlock the power of the disc method to calculate volumes of solids of revolution. Perfect your calculus skills with our comprehensive guide, from basic concepts to advanced applications.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026