Spherical dome surface distance formula

It’s useful to know the distance from the top of a dome to the foundation — the surface distance. The formula for our PCalc plugin works fine for spherical domes that are less than a hemisphere. Turns out it doesn’t work for spherical domes greater than a hemisphere. The formula would “invert” and only give the bottom portion surface distance.

Had to develop a new formula. We want the surface distance from the apex to the footer. We know the diameter and height of the dome. Using that we can find the angle from the apex to the footer. In any spherical dome that angle forms the “corner” of an isosceles triangle with the distance from the apex to the footer as the “base” and the radius of curvature forming the “sides” of the triangle. A little subtraction and multiplication and we have a formula for surface distance that works for any size spherical dome.

Too bad it’s not that easy for an elliptical dome.