Building a Cone
Suppose you want to build a (right circular) cone out of some flat material, perhaps paper or
metal. You cut out a sector of a circle and roll it up to make the cone. Let the radius of the
sector be s, its central angle T (in radians), the height of the cone be h, the radius of its base r,
and the vertex angle (i.e., the angle between its axis and any slantheight line) t (also in
radians).
You are given two of {h,r,s,t,T}, and wish to determine the other three. There are ten cases,
depending on what you are given:
 Case 1: You know h and r. Then
 t = Arctan(r/h),
 s = h/cos(t) = sqrt(h^{2}+r^{2}),
 T = 2*Pi*r/s.
 Case 2: You know h and t. Then
 r = h*tan(t),
 s = h/cos(t) = sqrt(h^{2}+r^{2}),
 T = 2*Pi*r/s = 2*Pi*sin(t).
 Case 3: You know h and s. Then
 r = sqrt(s^{2}h^{2}),
 t = Arccos(h/s),
 T = 2*Pi*r/s.
 Case 4: You know h and T. Then
 r = h*T/sqrt(4*Pi^{2}T^{2}),
 s = 2*Pi*r/T,
 t = Arctan(r/h).
 Case 5: You know r and s. Then
 T = 2*Pi*r/s,
 h = r*sqrt(4*Pi^{2}T^{2})/T,
 t = Arctan(r/h).
 Case 6: You know r and t. Then
 h = r*cot(t),
 s = r/sin(t),
 T = 2*Pi*r/s.
 Case 7: You know r and T. Then
 s = 2*Pi*r/T,
 h = sqrt(s^{2}r^{2}),
 t = Arctan(r/h).
 Case 8: You know s and t. Then
 h = s*cos(t).
 r = s*sin(t),
 T = 2*Pi*sin(t),
 Case 9: You know s and T. Then
 r = s*T/(2*Pi),
 h = sqrt(s^{2}r^{2}),
 t = Arctan(r/h).
 Case 10: You know t and T. Then the values of h, r, and s cannot be determined
without further information. You can determine their ratios h/s, r/s, and h/r as
follows:
 h/s = cos(t),
 r/s = sin(t) = T/(2*Pi),
 h/r = cot(t).
