# Sphere inscribed in a cone

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?)

• This isn't a homework Jan 19, 2019 at 12:08
• Can you made an image please? Jan 19, 2019 at 12:09
• Actually this considered the image to large, sorry Jan 21, 2019 at 13:17

## 2 Answers

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 $$OC$$?

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.

• Sorry to be so late, I appreciate your answer. Jan 8, 2020 at 14:58