Section 8.3: Volumes of Spheres

Property

The volume of a sphere is $\frac{2}{3}$ times the volume of a cylinder that has the same diameter as the sphere and height equal to the diameter:

When $h = 2r$ (diameter), this becomes:

Property

The volume of a sphere is given by

where $r$ is the radius of the sphere. Recall that $r^3$, which we read as '$r$ cubed,' means $r \times r \times r$.

Examples

• A gumball has a radius of 1 centimeter. Its volume is $V = \frac{4}{3} \pi (1)^3 = \frac{4}{3}\pi \approx 4.19$ cubic centimeters.
• A soccer ball has a diameter of 22 cm, so its radius is 11 cm. Its volume is $V = \frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) \approx 5575.28$ cubic centimeters.
• A spherical ornament has a volume of $36\pi$ cubic inches. To find its radius, solve $36\pi = \frac{4}{3}\pi r^3$, which simplifies to $27 = r^3$, so the radius is $r = \sqrt[3]{27} = 3$ inches.

Explanation

Volume measures the space inside a 3D shape, like a ball or a planet. For a sphere, you cube the radius (multiply it by itself three times), then multiply by pi ($\pi$), and finally multiply by the fraction $\frac{4}{3}$.

Property

The volume of a sphere can be derived by dividing it into many small pyramids with vertices at the center:

Examples