Pappus' theorem for surface area states that the surface area of a solid of revolution equals which product?

Pappus' theorem for surface area states that the surface area of a solid of revolution equals which product?

• A. Length multiplied by distance travelled
• B. Generating area multiplied by centroid distance travelled
• C. Radius multiplied by arc length
• D. Height multiplied by thickness

Answer: (D)

The statement being tested is Pappus’s second theorem for surface area: the surface area generated by revolving a plane curve about an external axis equals the length of the curve times the distance the curve’s centroid travels during the rotation. As the curve spins, its centroid traces a circle with radius equal to its distance from the axis, so the distance traveled by the centroid is the circumference of that circle, 2π times that radius. Thus the surface area equals the curve length times this centroid-path length. In other words, you multiply how long the generating curve is by how far its centroid moves. If the centroid is a distance d from the axis, the surface area is S = L × 2πd. The other options don’t capture this relationship: they mix in quantities that aren’t what Pappus’s theorem uses (such as height or radius in the wrong context, or an area rather than a curve length).

The statement being tested is Pappus’s second theorem for surface area: the surface area generated by revolving a plane curve about an external axis equals the length of the curve times the distance the curve’s centroid travels during the rotation. As the curve spins, its centroid traces a circle with radius equal to its distance from the axis, so the distance traveled by the centroid is the circumference of that circle, 2π times that radius. Thus the surface area equals the curve length times this centroid-path length.

In other words, you multiply how long the generating curve is by how far its centroid moves. If the centroid is a distance d from the axis, the surface area is S = L × 2πd. The other options don’t capture this relationship: they mix in quantities that aren’t what Pappus’s theorem uses (such as height or radius in the wrong context, or an area rather than a curve length).