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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?)

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

2 Answers 2

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

enter image description here

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 $OC$?

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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.

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

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