How to Find the Surface Area of a Torus

Calculates the surface area of a torus as 4 × π² × center distance × tube radius, by the theorem of Pappus. It is the circumference of the tube, 2πr, swept along the circle of length 2πR that the tube follows.

Explanation

A torus is the doughnut shape a circle sweeps out as it turns about an axis outside it. Its surface is what the rim of the tube traces as it goes around.

S = 4\pi^2 R r

$S$ — the surface area
$R$ — the center distance, from the center of the doughnut to the center of the tube
$r$ — the radius of the tube

$R$ must be greater than $r$ .

The theorem of Pappus

Take the length of the curve that was turned and multiply it by the distance its centroid travelled. The cross-section of the tube has circumference $2\pi r$ , and its center travels $2\pi R$ .

S = 2\pi r \times 2\pi R = 4\pi^2 R r

The volume uses the area of the cross-section, $\pi r^2$ ; the surface area uses its perimeter, $2\pi r$ . That is the only difference between the two formulas.

Example

With the defaults, a center distance of $R = 10$ and a tube radius of $r = 3$ .

S = 4\pi^2 \times 10 \times 3 = 120\pi^2 \approx 1184.3525

A curiosity

The surface area depends only on the product $Rr$ . A torus with $R = 10, r = 3$ and one with $R = 6, r = 5$ both have $Rr = 30$ , so their surfaces are exactly the same size. Their volumes are not: $2\pi^2 R r^2$ squares the tube radius, so the fatter tube holds more.