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?)
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$\begingroup$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 $OC$?
$\endgroup$ $\begingroup$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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