How do I find the surface area of a Torus using calculus?

Question

I managed to find the volume of a torus by solids of revolution, using the washers method. Can the same method be applied for finding an expression for the surface area?

Answer 1

Choosing a toroidal coordinate system the computation of area is I believe most easy. However, a hybrid between spherical and cylindrical coordinate system can be adopted.

Let radius at crown be $b$, $R$ tube radius, $r$ variable radius of cyl coordinate system, and $\phi$ the latitude of spherical system. Area of torus =

$$ \int_{b-R}^{b+R} \int_{-\pi}^{ \pi} R\, d\phi \cdot 2 \pi\, dr = 2 \pi R \int_{-\pi} ^{ \pi} ( b- R \cos \phi) d \phi = 2 \pi R \cdot 2 \pi b. $$

Answer 2

One can find the area, using an analogous of the washers method, as follows.

If $O$ is the torus center and $C$ the center of the revolving circle, consider a point $P$ on that circle corresponding to an angle $\theta=\angle OCP$, and a point $P'$ close to $P$, corresponding to an angle $\theta+d\theta$. If $R=OC$ and $r=CP$, then the area of the stripe between the two circles obtained by the rotation of $P$ and $P'$ is: $$dA=2\pi\cdot PH\cdot PP' = 2\pi(R-r\cos\theta)(r\,d\theta).$$ Integrate that between $0$ and $2\pi$ and you are done.