Volume of a Sphere
In short: The volume of a sphere is the space inside a perfectly round ball, found from its radius with the formula V = (4/3)πr³.
8th GradeGeometry8.G.C.9
What is Volume of a Sphere?
The volume of a sphere tells how much a ball can hold, where a sphere is every point at a fixed distance, the radius \(r\), from a center, and it grows very quickly: doubling the radius makes the volume 8 times larger.
The formula is \(V = \frac{4}{3}\pi r^3\). The radius is cubed, not squared, because volume is measured in cubic units. Cubing happens before multiplying by \(\frac{4}{3}\pi\).
How to spot it
- The object is a ball, globe, bubble, marble, planet or any perfectly round solid.
- A diameter is given and the question needs capacity or how much air fits inside.
- Two spheres are compared and one radius is a multiple of the other.
- A hemisphere is mentioned, so you will compute a whole sphere and take half.
The method
- Find the radius, halving the diameter if that is what you are given.
- Cube the radius: multiply it by itself three times.
- Multiply the cube by \(\frac{4}{3}\pi\), keeping π as a symbol or using 3.14159.
- Write cubic units, and round only at the end.
Volume of a Sphere: solved example
Problem
A decorative glass ball in a Santa Fe gallery has a radius of 7.5 centimeters. What is its volume to the nearest tenth of a cubic centimeter?
Solution
Cube the radius: \(7.5^3 = 421.875\). Multiply by \(\frac{4}{3}\): \(421.875 \times \frac{4}{3} = 562.5\), so \(V = 562.5\pi\).
With \(\pi \approx 3.14159\), \(562.5 \times 3.14159 \approx 1{,}767.1\). The ball has a volume of about 1,767.1 cm³.
Try it
Question 1
Give the volume of a sphere with radius 3 inches in terms of π.
Show the answer
\(\frac{4}{3}\pi \cdot 27 = 36\pi\) cubic inches, about 113.1.
Question 2
A sphere has volume \(\frac{32\pi}{3}\) cm³. Find its radius.
Show the answer
\(\frac{4}{3}\pi r^3 = \frac{32\pi}{3}\) gives \(r^3 = 8\), so \(r = 2\) cm.
Volume of a Sphere vs. Surface Area
Surface area is the skin of the ball, measured in square units, \(4\pi r^2\). Volume is the inside, measured in cubic units, \(\frac{4}{3}\pi r^3\). The exponent tells you which is which: 2 for the outside, 3 for the inside.
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