Sphere inscribed in a cone

Question

If a cone of height h and radius r has a sphere inscribed in it such that it touches the base and the curved surface area, how can I find the radius of the sphere? (Is this in the level of a 9 grader?)

Answer 1

Is the cone a right-circular cone? Consider the following cross-section picture.

There are two similar triangles in the picture: $\bigtriangleup ABD$ and $\bigtriangleup COD$.

Use the property of similar triangles and form the following: $$\frac{BD}{BA}=\frac{OD}{OC}$$

Note that $AB$ is the radius of the cone. $OD$ is equal to the height of the cone minus the radius of the sphere.$OC$ is the radius of the sphere.

Question for you: can you write $\frac{BD}{BA}=\frac{OD}{OC}$ in terms of $h$ and $r$ and solve for $OC$?

Answer 2

Hint: Consider the cross-sectional diagram (i.e. What does the object look like if I slice it down the middle?)

You should end up with an isosceles triangle with a circle inscribed.