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Michael Lee's Mathematics and Physics

Surface area of a torus and its portions or regions by integration.  Derivation.


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 or portions thereof.  All angle measurements are in radians.




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.  For a ring torus R is either greater than or equal to r.






This is a cross section of the tube as if it had been cut vertically on the left side of the overview above.  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 turning right.





Here is a diagram for the cross section shown above.  Angle theta is shown in a sample position and points to ds; it can range from negative PI / 2  to positive PI / 2.    Notice how segment X indicates how far away ds is from the centre of the torus.  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 a portion or partial areas of a torus. 


Suppose we wish to find the surface area of a portion of the outer surface of a torus as shown in the following cross-sectional diagram where...




So how might this be done?  Well you just take our formula from above for the outer surface and integrate that region as follows:


                                         For the outer side only.  




                The formula for the inner side can be derived in a similar fashion. 

                                         For the inner side only