Question

Drag the points to create two different cylinders with the same volume. You can change both the height and radius of each cylinder.



What is the volume of the cylinders?

Answer 1

The volume \( V \) of a cylinder can be calculated using the formula:

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

where \( r \) is the radius of the cylinder's base and \( h \) is the height of the cylinder.

To create two different cylinders with the same volume, you can choose different pairs of values for the radius \( r \) and height \( h \) such that the volumes calculated using the formula are equal.

For example:

1. **Cylinder 1:**
- Height \( h_1 = 5 \) units
- Radius \( r_1 = 2 \) units
- Volume:
\[
V_1 = \pi (2^2)(5) = \pi (4)(5) = 20\pi
\]

2. **Cylinder 2:**
- Height \( h_2 = 4 \) units
- Radius \( r_2 = \sqrt{\frac{20}{4\pi}} \approx 2.24 \) units (this can be derived by setting the volumes equal)
- Volume:
\[
V_2 = \pi (r_2^2)(4) = \pi \left(\sqrt{\frac{20}{4\pi}}\right)^2 (4) = \pi \left(\frac{20}{4\pi}\right)(4) = 20\pi
\]

In this case, both cylinders have the same volume of \( 20\pi \) cubic units, but different radii and heights.

You can choose any other combinations of \( r \) and \( h \) as long as \( V_1 = V_2 \).