Problem 22 - A History of Mathematics: An Introduction

Question

Find the volume of a torus by applying Pappus's theorem. Assume that the torus is formed by revolving the disk of radius \(r\) around an axis whose distance from the center of the disk is \(R>r\).

Step-by-Step Solution

Answer

Answer: The volume of the torus is given by the formula \(V = 2\pi^2r^2R\).

Step 1: Find the area of the revolving plane figure (the disk)

The revolving plane figure is a disk of radius \(r\). The area of a disk can be found using the formula \(A = \pi r^2\). In this case, \(A = \pi r^2\).

Step 2: Find the distance traveled by the centroid of the disk upon revolving a full revolution

The centroid of the disk is located at the center. As the disk revolves around the axis, its centroid will trace a circle with radius \(R\). The circumference of the circle is given by the formula \(C = 2\pi R\). So, the distance traveled by the centroid upon revolving a full revolution is \(C = 2\pi R\).

Step 3: Apply Pappus's theorem to find the volume of the torus

According to Pappus's theorem, the volume of the solid of revolution (the torus) is equal to the product of the area of the revolving plane figure (the disk) and the distance traveled by its centroid upon revolving (the circumference of the circle traced by the centroid). Thus, the volume \(V\) of the torus is given by the formula \(V = AC\). Substitute the area \(A = \pi r^2\) and the distance \(C = 2\pi R\) into the formula: \(V = (\pi r^2)(2\pi R) = 2\pi^2r^2R\).

Step 4: Final answer for the volume of the torus

The volume of the torus formed by revolving the disk of radius \(r\) around an axis whose distance from the center of the disk is \(R > r\) is: \(V = 2\pi^2r^2R\).

Key Concepts

Volume of a Torus

A torus is a donut-shaped figure, which can be visualized as being formed by revolving a smaller circle (in this case, a disk) around an axis that lies outside the circle. The volume of a torus is important in both mathematics and physics and can be elegantly found using Pappus's theorem. Pappus's theorem relates the volume of a solid of revolution to the centroid of the shape being revolved.
To find the volume of a torus, first consider the smaller circle, a disk of radius \(r\), which revolves around an axis placed at a distance \(R\) from the center of the disk (where \(R > r\)). The formula for the volume of the torus using Pappus's theorem is

• Volume \(V = 2\pi^2r^2R\)

This formula arises from multiplying the area of the disk \(A = \pi r^2\) by the path length that the centroid of the disk travels in one full circle, \(C = 2\pi R\). This gives the final volume as \(V = 2\pi^2r^2R\), showcasing how Pappus's theorem simplifies the calculation.

Solid of Revolution

A solid of revolution is a three-dimensional shape that is formed when a two-dimensional shape is rotated around an axis. This is a key concept in calculus and geometry, particularly when calculating volumes of complex shapes. For instance, by revolving a disk about an external axis, you create a torus.
The process involves considering the area of the original shape—it could be a circle, rectangle, or another shape—and understanding the path its centroid travels during one complete revolution. This path defines the outer profile of the solid of revolution. In the case of a torus, as the disk of radius \(r\) is rotated around an axis at a distance \(R\), every point on the disk travels in a circular path, creating the toroidal shape.
Applying Pappus's theorem provides a powerful technique to determine the volume of these interesting shapes, by focusing on the geometric properties of the motion.

Centroid of a Disk

To fully appreciate Pappus's theorem, it's important to understand the concept of the centroid. The centroid of a shape is effectively its "center of mass," assuming uniform density. For a disk, the centroid is located at the center of the circle.
When calculating the volume of a solid of revolution, like a torus, using Pappus's theorem, the movement of the centroid is crucial. It determines the trajectory traced during the rotation. For a disk of radius \(r\), revolving around an axis at distance \(R\), the centroid moves in a circle with a radius \(R\). This movement path forms part of the basis for determining the length of the path that contributes to the volume formula.

• Centroid movement path = \(2\pi R\)

The location and path of the centroid provide the necessary measurements to apply Pappus's theorem effectively, turning sometimes complex volume calculations into more intuitive geometric exercises.