# Thread: Inscribed Spheres

1. ## Inscribed Spheres

A cone with a 10-inch diameter is 12 inches tall.
a. Find the radius of the largest sphere that can be inscribed in the cone
b. The volume of this sphere is what percentage of the volume of the cone?

Is there an easy way to do this with any shapes inside any shapes?

2. Hello, stones44!

A cone with a 10-inch diameter is 12 inches tall.

a. Find the radius of the largest sphere that can be inscribed in the cone.

b. The volume of this sphere is what percentage of the volume of the cone?
Code:
    - : - - -  5  - - - D - - -  5  - - - :
C * - - - - - - - * * * - - - - - - - * B
:  \          *     |     *          /
:   \       *       |       *       /
:    \     *        |r       *     /
:     \             |             /
:      \  *         |         *  /
:       \ *        O*         * /
:        \*         | \       */
12         \         |   \r    /
:          *        |     \  *
:           *       |       *E
:            \*     |     */
:             \   * * *   /
:              \    |    /
:               \   |   /
:                \  |  /
:                 \ | /
:                  \|/
-                   *
A

We have: . $DB = 5,\:DA = 12$
. . From Pythagorus: . $DA = 13$

The center of the sphere is $O.$
Radii $OD = OE = r$
. . Hence: . $PA = 12-r$

We have similar right triangles: . $\Delta OEA \sim \Delta BDA$

So we have: . $\frac{OE}{OA} \:=\:\frac{BD}{AB}\quad\Rightarrow\quad\frac{r}{12-r} \:=\:\frac{5}{13}$

Therefore: . $\boxed{r \:=\:\frac{10}{3}}$
.

3. ehh im a little confused..i think your diagram is missing variables..and how to you get the answer from the equation?