Building a Frustum

Suppose you want to build a frustum of a (right circular) cone out of some flat material, perhaps paper or metal. You cut out a sector of an annulus and roll it up to make the curved surface of the frustum. Let the major radius of the sector be S, its minor radius be (S-s), its central angle T (in radians), the height of the frustum be h, the radius of its base R, the radius of its top r, and the vertex angle (i.e., the angle between its axis and any slant-height line) t (also in radians).

You are given some of {h, R, r, S, s, t, T}, and wish to determine all the others. You need either three of the five lengths, or else two lengths and one of the two angles, to determine all of the others. There are thirty cases. If you know both angles and only one length, you can only determine the ratios of the unknown lengths.

The relationships among these parameters boil down to the following:

(R-r)/s = r/(S-s) = R/S = T/(2*Pi) = sin(t),

s² = h² +(R-r)²,

R - r = h*tan(t),

s = h*sec(t).

A typical situation is when you are given the three lengths h, R, and r. Then

t = Arctan([R-r]/h),

sin(t) = (R-r)/sqrt(h² +[R-r]²),

S = R/sin(t),

s = S - r/sin(t),

T = 2*Pi*sin(t).

Another typical situation is when you are given the two lengths S and s, and angle T. Then

t = Arcsin(T/[2*Pi]),

r = (S-s)*T/(2*Pi),

R = S*T/(2*Pi),

h = s*cos(t).