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Please see this diagram: https://www.dropbox.com/s/73eics7x43bgiwm/surface-area-torus.png?dl=0

This is a cross section through a torus, the dashed line is the central axis. I am interested in the external surface area of the region marked with the solid black line.

A few notes about the diagram: The angle alpha is between π/2 and π radians, and the vertical line of the sector is parallel to the axis of the torus (dashed line on the diagram).

Given the simple geometry, is there a more simple/direct approach than a long process of integration to determine this area? Is there a known expression I can use here?

Any help would be welcome.