Pappus’s Theorem Applications — Theory
1. What is Pappus’s centroid theorem?
Pappus’s centroid theorem connects geometry, centroids, and solids of revolution. It says that a
volume or surface area of revolution can often be found without doing the full integration.
The key idea is simple: multiply the original measure by the distance traveled by its centroid.
2. Volume theorem
If a plane region has area \(A\), and it is rotated about an external axis in the same plane, then
the generated volume is
\[
V=A(2\pi d),
\]
where \(d\) is the distance from the centroid of the region to the axis of rotation.
\[
d=\operatorname{dist}(\bar C,\text{axis}).
\]
3. Surface area theorem
If a plane curve has length \(L\), and it is rotated about an external axis in the same plane, then
the generated surface area is
\[
S=L(2\pi d),
\]
where \(d\) is the distance from the centroid of the curve to the axis of rotation.
4. Distance from the centroid to the axis
If the axis is horizontal,
\[
y=c,
\]
then the centroid distance is
\[
d=|\bar y-c|.
\]
If the axis is vertical,
\[
x=c,
\]
then the centroid distance is
\[
d=|\bar x-c|.
\]
5. Why the theorem works
For volume, a small area element \(dA\) at distance \(r\) from the axis sweeps out approximately
a shell volume:
\[
dV=2\pi r\,dA.
\]
Therefore,
\[
V=2\pi\iint_\Omega r\,dA.
\]
If the axis does not intersect the region, the signed distance has one direction. The average distance
is the centroid distance \(d\), so
\[
V=2\pi A d.
\]
6. Direct integration comparison
Pappus’s theorem gives the same result as a direct integral when the hypotheses are satisfied.
For volume,
\[
V=A(2\pi d)
=
2\pi\iint_\Omega r\,dA.
\]
For surface area,
\[
S=L(2\pi d)
=
2\pi\int_C r\,ds.
\]
The calculator includes a quick numerical check using these equivalent direct-integral forms.
7. Worked example: disk rotated about an external axis
Suppose a disk of radius \(R\) is rotated about an axis in its plane. The center of the disk is
distance \(d\) from the axis, and the axis does not intersect the disk.
The area of the disk is
\[
A=\pi R^2.
\]
The centroid is the center of the disk, so the distance traveled by the centroid is
\[
2\pi d.
\]
Therefore, the volume is
\[
V=A(2\pi d)
=
\pi R^2(2\pi d)
=
2\pi^2R^2d.
\]
This is the standard torus volume formula.
8. Worked example: circle curve rotated about an external axis
A circle curve of radius \(R\) has length
\[
L=2\pi R.
\]
If its center is distance \(d\) from the axis of rotation, Pappus’s surface theorem gives
\[
S=L(2\pi d)
=
(2\pi R)(2\pi d)
=
4\pi^2Rd.
\]
This is the surface area of a torus.
9. Common centroids used with Pappus’s theorem
10. When Pappus’s theorem cannot be used directly
The usual form of Pappus’s theorem requires the rotation axis to be external. This means the axis must
not intersect the region or curve being rotated.
If the axis passes through the object, direct integration is usually needed instead. In that case,
the swept distance is not simply \(2\pi d\) for every part of the object.
11. Common mistakes
- Using the wrong centroid: region centroid and curve centroid are not always the same.
- Using area for surface area: surface theorem uses curve length \(L\), not area \(A\).
- Using length for volume: volume theorem uses area \(A\), not curve length \(L\).
- Letting the axis intersect the object: the standard theorem requires an external axis.
- Forgetting the full centroid path: the centroid travels \(2\pi d\), not just \(d\).
