Comparing a sphere and a cylinder
When a problem asks you to compare volumes, first identify the radius and any height given for the cylinder. Use the volume formulas:
- Sphere: (V=\frac{4}{3}\pi r^3)
- Cylinder: (V=\pi r^2 h)
Method
- Read the dimensions carefully. Make sure the sphere and cylinder use the same radius when that is intended.
- Substitute into each formula. Keep (\pi) as long as possible for an exact answer.
- Simplify each volume. Combine like factors and reduce fractions if needed.
- Compare the results. You may be asked for the difference, ratio, or which volume is larger.
Useful comparison idea
If the cylinder has the same radius as the sphere and height (h), then comparing often comes down to comparing (\frac{4}{3}r^3) with (r^2h). After factoring out common terms, the calculation becomes easier.
Check your work
- Verify that the units are cubic units.
- Recompute any arithmetic with the powers of the radius.
- If the answer should be exact, leave it in simplest form with (\pi) unless the instructions say otherwise.