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 slant-height 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(h²+r²),

T = 2*Pi*r/s.

Case 2: You know h and t. Then

r = h*tan(t),

s = h/cos(t) = sqrt(h²+r²),

T = 2*Pi*r/s = 2*Pi*sin(t).

Case 3: You know h and s. Then

r = sqrt(s²-h²),

t = Arccos(h/s),

T = 2*Pi*r/s.

Case 4: You know h and T. Then

r = h*T/sqrt(4*Pi²-T²),

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*Pi²-T²)/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(s²-r²),

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(s²-r²),

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).