How to Calculate the Volume of a Sphere
Blow up a balloon until its radius doubles. How much more air is inside? Not twice as much. Eight times as much.
That's because volume grows with the cube of the radius. It's the key fact behind the sphere volume formula, and it explains everything from why large raindrops fall faster to why a planet only slightly larger than Earth would be far more massive.
Here's how to calculate a sphere's volume, step by step.
A Sphere Fills Exactly Two-Thirds of Its Box
Put a ball snugly inside a cylinder that's exactly as tall and wide as the ball. What fraction of the cylinder does the ball fill?
Exactly two-thirds. Not approximately. Precisely 2/3, for any size sphere. Archimedes proved this around 250 BC, and he was so proud of it that he asked for a sphere inside a cylinder to be carved on his tomb.
That ratio leads directly to the formula.
The Formula
V = (4/3) × π × r³
- V is the volume
- r is the radius
- π ≈ 3.14159
The result is in cubic units: cm³, m³, cubic inches.
Where it comes from: a cylinder of radius r and height 2r has volume πr² × 2r = 2πr³. Two-thirds of that is (4/3)πr³. See the geometry formulas.
Example 1: From the Radius
A ball has a radius of 5 cm.
r³ = 5 × 5 × 5 = 125
V = (4/3) × π × 125 ≈ 523.60 cm³
Look up cubes in the cube numbers list.
Example 2: From the Diameter
A basketball has a diameter of about 24 cm.
Option A: halve it first. r = 12 cm.
V = (4/3) × π × 12³ = (4/3) × π × 1,728 ≈ 7,238 cm³
Option B: use the diameter formula directly:
V = π × d³ / 6 = π × 24³ / 6 = π × 13,824 / 6 ≈ 7,238 cm³
That's about 7.24 liters of air, since 1,000 cm³ = 1 liter. Convert with the volume unit converter.
Example 3: Planet Earth
Earth's mean radius is about 6,371 km.
V = (4/3) × π × 6,371³ ≈ 1.083 × 10¹² km³
That's about 1.08 trillion cubic kilometers. (Earth is slightly flattened at the poles, so this is an approximation using the mean radius.)
Working Backward: From Volume to Radius
Rearrange the formula:
r = ∛(3V / (4π))
A spherical tank holds 1,000 liters (1 m³):
r = ∛(3 × 1 / (4π)) = ∛0.2387 ≈ 0.620 m
So the tank's diameter is about 1.24 meters. Use the scientific calculator for cube roots.
Why Doubling the Radius Gives 8× the Volume
When every length is multiplied by k, volume is multiplied by k³:
| Radius multiplied by | Volume multiplied by |
|---|---|
| 2 | 8 |
| 3 | 27 |
| 10 | 1,000 |
| 0.5 | 0.125 |
This square–cube law has real consequences. A raindrop's weight grows with its volume (r³), but air resistance grows with its cross-sectional area (r²), so bigger drops fall faster.
Related: Surface Area
The surface area of a sphere is:
S = 4πr²
For r = 5 cm: S = 4 × π × 25 ≈ 314.16 cm². Notice it's exactly the derivative of the volume formula with respect to r: growing a sphere by a thin layer adds volume equal to its surface area times the thickness.
An Insider Reference: Zu Geng and Cavalieri's Principle
Archimedes wasn't the only one to crack the sphere. In 5th–6th century China, mathematician Zu Geng, son of the famous astronomer Zu Chongzhi, derived the sphere's volume using a principle stated roughly as: if two solids have equal cross-sectional areas at every height, they have equal volumes.
More than a thousand years later, Italian mathematician Bonaventura Cavalieri published the same idea in 1635. It's now called Cavalieri's principle. Using it, you can show that a hemisphere has the same volume as a cylinder with a cone removed from it, which again gives (2/3)πr³ for the hemisphere and (4/3)πr³ for the full sphere.
Common Mistakes
1. Using the diameter as r. This makes the answer 8 times too big.
2. Squaring instead of cubing. Volume uses r³; surface area uses r².
3. Forgetting the 4/3. Multiply by 4 and divide by 3, or multiply by about 4.18879 in one step.
4. Unit confusion. A radius in centimeters gives cm³. 1 m³ = 1,000,000 cm³.
Two Concepts Worth Knowing
Cube
The cube of a number is that number multiplied by itself three times: 5³ = 125. Cubing is why volume scales so quickly with size.
Cavalieri's Principle
Cavalieri's principle says solids with equal cross-sections at every height have equal volumes. It's a forerunner of integral calculus. See the calculus formulas.
Quick Answer: How Do You Calculate the Volume of a Sphere?
Use V = (4/3)πr³, where r is the radius. For r = 5 cm, V = (4/3) × π × 125 ≈ 523.6 cm³. If you know the diameter d, use V = πd³/6. To find the radius from the volume, use r = ∛(3V/4π).
Try Them Yourself
- Geometry Formulas: sphere volume and surface area
- Cube Numbers List: quick r³ values
- Volume Unit Converter: cm³, liters and gallons
- Scientific Calculator: cube roots for working backward
- Archimedes: The Mathematician Who Was Centuries Ahead of His Time: the proof behind the formula
- How to Calculate the Area of a Circle: the 2D version
Measure the circumference of a ball with a string, calculate its radius (r = C / 2π), then its volume. Fill a measuring jug with water to compare, if the ball can be submerged.