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                                      Michael Lee's Mathematics and Physics

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


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.




Here is a view of the curvature of a torus.  The outer surface is said to be positively curved whereas the inner surface is said to be negatively curved.  What this means is, if a tangent plane is placed on any point on the outer surface (positively curved), all the points on the surface of the torus will be curving away from it on one side of the plane only, whereas a tangent plane located on any point on the inner surface (negatively curved), results of points on the torus  that curve away on both sides of the plane.




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 (negative curvature) is on the right and the outer side (positive curvature) is on the left.  It's like riding through the tubes in the London Underground turning right while travelling forward.





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 (negative curvature), it is the Major Radius less line segment x or (R-x) and for the outer side (positive curvature), it is the Major Radius plus line segment x or (R+x) owing to its symmetry.  The differential ds is then spun around the surface of the torus.




 The Inner Surface or Negative Curvature


The Outer Surface or Positive Curvature



How much larger is the outer surface (positive curvature) than the inner surface (negative curvature)?







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 



The surface area of a frustum of a torus.
Here is a cross section of the frustum of a torus; it's the yellow portion of the torus.