A circular cylinder of radius r is inscribed in a sphere of radius of 2r Find a formula for the volume of           

We are asked to find the volume of a circular cylinder of radius \( r \) that is inscribed in a sphere of radius \( 2r \).

### Step 1: Geometry of the problem

The cylinder is inscribed in the sphere, which means that the cylinder is entirely contained within the sphere, and the diameter of the sphere will be equal to the diagonal of the cylinder. 

- The radius of the sphere is \( 2r \), so the diameter of the sphere is \( 4r \).
- The radius of the base of the cylinder is \( r \).
- Let the height of the cylinder be \( h \).

By using the Pythagorean theorem, we can relate the height \( h \) of the cylinder to the radius of the sphere. Since the diagonal of the cylinder is the diameter of the sphere, we have:

\[
\text{Diagonal of the cylinder} = \sqrt{h^2 + (2r)^2} = 4r
\]

### Step 2: Solving for the height \( h \)

Now, using the relationship for the diagonal:

\[
\sqrt{h^2 + (2r)^2} = 4r
\]

Square both sides:

\[
h^2 + (2r)^2 = (4r)^2
\]

Simplifying:

\[
h^2 + 4r^2 = 16r^2
\]

\[
h^2 = 12r^2
\]

\[
h = \sqrt{12r^2} = 2\sqrt{3}r
\]

### Step 3: Volume of the cylinder

The volume \( V \) of a cylinder is given by the formula:

\[
V = \pi r^2 h
\]

Substitute \( h = 2\sqrt{3}r \) into the volume formula:

\[
V = \pi r^2 (2\sqrt{3}r)
\]

Simplifying:

\[
V = 2\sqrt{3} \pi r^3
\]

### Final Answer:

The volume of the inscribed cylinder is:

\[
V = 2\sqrt{3} \pi r^3
\]

Would you like further details or clarifications?

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Here are 5 related questions you could explore:
1. What is the surface area of the cylinder inscribed in the sphere?
2. How does the volume change if the radius of the sphere increases?
3. What is the ratio of the volume of the cylinder to the volume of the sphere?
4. Can you derive a general formula for the volume of a cylinder inscribed in a sphere of any radius?
5. How would the solution change if the cylinder had an elliptical base?

**Tip:** Always check the geometric relationships between shapes when they are inscribed in one another, as they provide key insights for solving the problem.

A circular cylinder of radius r is inscribed in a sphere of radius 2r. Find a formula for the volume of the cylinder.

Discover how to calculate the volume of a cylinder with radius r inscribed in a sphere of radius 2r. This problem uses the Pythagorean theorem and the formula for the volume of a cylinder. Suitable for students in Grades 9-12.

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