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A right circular cone has radius 8 cm and height 20 cm. A sphere is inscribed in the cone such that it touches the base and the lateral surface. Find the radius of the inscribed sphere.

1. 3.35 cm
2. 5.42 cm
3. 2.8 cm
4. 4.21 cm

Correct answer: 5.42 cm

Solution

The axial section of the cone is an isosceles triangle with base \(16\) cm, equal sides \(\sqrt{8^2+20^2}=\sqrt{464}\), and height 20 cm. The inradius of this triangle equals the radius of the inscribed sphere, and using the triangle inradius formula gives approximately 5.42 cm. Hence the sphere’s radius is 5.42 cm.

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