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Surface Area of a Torus

 

This is what a torus looks like; it's a donut shaped body.  Keep this shape in mind when thinking about the problem of calculating its surface area.  Notice how it is composed of "panels."  The sizes of the panels can be calculated by restricting the range of the integral equation below and then cut the torus up like it were a pizza.

 

 

 

This is a cross section of the tube as if it had been cut vertically on the left side.  The inner side is on the right and the outer side is on the left.  It's like riding through the tubes in the London Underground.

 

Overview of a Torus.  Adding some dimensions R (the major radius) and r (the minor radius).   R is the distance from the centre of the torus to the centre of the tube and r is the radius of the tube.

 

 

 

 

 

Here is a diagram for the cross section shown above.  Angle theta is shown in a sample position and points to ds.   Notice how segment X indicates how far away ds is from the centre.  For the inner side, it is the Major Radius less line segment X and for the outer side, it is the Major Radius plus line segment X owing to its symmetry.

 

 

  

 

 

 

 

The surface area of each panel can be calculated by adjusting the range of the integral equation and then, from the overview (see drawing above), cut the torus up like a pizza.