Sphere inscribed in a cone

Many problems about inscribed solids are solved by passing to a plane section: the sphere inscribed in a cone is a typical example.

Example — Sphere inscribed in a cone

In a right cone, the inscribed sphere has its centre on the axis of the cone. Sectioning the cone with a plane passing through the axis gives an isosceles triangle, and the section of the sphere is the circle inscribed in that triangle. The radius of the sphere is then obtained as $r_{\text{sfera}} = \frac{S_{\text{triangolo}}}{s},$ where $s$ is the semiperimeter of the section triangle.

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Topics: Synthetic geometry of space
Concepts: Cone · Sphere
Skills: Calculating · Synthetic geometry