Find the radius of the largest sphere that can inscribed in a right circular cone with height H and?

View the figure in profile. You will see an equivalent problem in two dimensions. Given an isosceles triangle with base 2R and corresponding height H, find the inradius.

Let the inradius be r.

area of triangle = HR

length of a leg = √(H² + R²)

perimeter = 2R + 2√(H² + R²)

The area of any triangle is half the inradius times the perimeter.

(1/2)r[2R + 2√(H² + R²)] = HR

r[R + √(H² + R²)] = HR

r = HR / [R + √(H² + R²)]

r = H / [1 + √(1 + H²/R²)]

For a cone of arbitrary height H and base radius R, the radius of the inscribed circle/sphere is:

r = H/(1+√(1+(H/R)²))

so that the volume of the sphere is (4/3)πr³.

If you don't set constraints over H, r or both the problem has no solution as the radius of the sphere increases as r increases