$\begingroup$

I know that you can use the formulas described in Pappus' centroid theorem, detailed here. But does Pappus' centroid theorem hold true for all forms of a torus: ring, horn, and spindle? I found another website that uses Pappus' centroid theorem for the volume and surface area of a ring or horn torus, but a different formula for the spindle torus.

This is the formula listed for the spindle torus:$$ V = \frac23 \pi ( 2r^2 + R^2 ) \sqrt{r^2 - R^2} + \pi r^2 R \left[\pi + 2\arctan\left( \frac{R}{\sqrt{r^2 - R^2}} \right) \right] $$where $r$ is the minor radius and $R$ is the major radius.

Is this formula for the spindle torus' volume accurate, or do I just have to use Pappus' centroid theorem?

$\endgroup$

1 Answer

$\begingroup$

Pappus's Centroid Theorem assumes that the solid generated does not intersect itself so I would use the formula you listed since a spindle torus does intersect itself.

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy