What is the simplest way to calculate the surface area of a region of a torus? 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.