How to Find the Volume of a Sphere

A sphere is the only common solid that needs just one measurement. No height, no base, no slant. Cube the radius and the rest is two constants. That simplicity is also the trap, because cubing punishes a sloppy radius three times over.

One measurement, cubed

Every length in a sphere is the radius in disguise, so the volume can only depend on it. Since volume must scale as a length cubed, the formula has to look like some number times r³, and the only real question is what that number is. It turns out to be 4π over 3, which is about 4.1888.

So a sphere holds a little over four times its radius cubed. That is a useful sanity check to carry: for a radius of 6, r³ is 216, and the answer should land somewhere near 900. It does, at 904.78.

Two thirds of its cylinder

Here is the check worth knowing, and it is old enough to have a biography. Take the smallest cylinder a sphere will fit inside: same radius, and a height of two radii. The sphere occupies exactly two thirds of it. Not approximately, exactly, for every sphere there has ever been.

Run the numbers. The cylinder has radius 6 and height 12, so it holds π times 36 times 12, which is 1357.17. Two thirds of that is 904.78, matching the formula to the last decimal. The same two thirds holds for the surfaces as well: the sphere’s 452.39 against the cylinder’s total of 678.58.

The leftover third has a tidy identity too. Subtract the sphere from the cylinder and you are left with 452.39, which is exactly two cones of radius 6 and height 6. Cylinder, sphere and cone at matching dimensions come in the ratio 3 to 2 to 1, which is the cleanest fact in solid geometry and worth more than any formula on this page.

A worked check you can repeat on paper

Two rows show 452.39 for different reasons, which is a coincidence of this radius rather than a rule. The leftover space in the cylinder happens to equal the sphere’s surface area only because r is 6. Change the radius and they part company immediately, so do not let it become a habit.

Cubing makes small errors large

The radius enters this formula three times, so any error in it is roughly tripled in the answer. Measure the radius one percent too big and the volume comes out three percent too big. That is the price of the formula’s simplicity, and it is why a diameter measured across the widest point beats a radius guessed from the centre.

Sphere	Radius	Volume
The worked example	6	904.78
Radius doubled	12	7238.23
Radius halved	3	113.10
A football, 22 cm across	11 cm	5575 cm³

Doubling the radius multiplies the volume by eight, not by two. Halving it leaves an eighth. That is why a ball twice the width of another holds eight times as much air, and why the football in the last row comes to 5.58 litres, which surprises most people who guess.

Mistakes that catch people out

• Using the diameter as the radius. The worst error on this page, because cubing turns a factor of two into a factor of eight. A diameter of 12 fed in as a radius gives 7238.23 instead of 904.78.
• Squaring instead of cubing. 4πr² is the surface area, not the volume. For this sphere that is 452.39, exactly half the right answer, which makes it look almost plausible.
• Dropping the four thirds. πr³ alone gives 678.58. The four thirds is not decoration, it is a quarter of the answer again.
• Rounding the radius early. One percent out on the radius is three percent out on the volume. Round at the end, never at the start.
• Answering in square units. Volume is cubic. Three lengths went into it, so three units come out.

Try these

Common questions

What is the formula for the volume of a sphere?

Four thirds times π times the radius cubed. A radius of 6 gives 4 over 3 times π times 216, which is 904.78 cubic units, or exactly 288π.

How do you find the volume from the diameter?

Halve it to get the radius, or use π d³ over 6 directly. A diameter of 12 gives π times 1728 over 6, which is 904.78, the same answer.

How do you find the radius from the volume?

Multiply the volume by 3, divide by 4π, then take the cube root. A volume of 904.78 gives 216, and the cube root of 216 is a radius of 6.

Why four thirds?

Because a sphere fills two thirds of the cylinder that just contains it. That cylinder holds 2πr³, and two thirds of 2πr³ is 4πr³ over 3. Archimedes established the two thirds without calculus, and everything else follows from it.

How much bigger is a sphere of twice the radius?

Eight times, because the radius is cubed. A sphere of radius 12 holds 7238.23 against 904.78 for one of radius 6. Its surface area only goes up fourfold, which is why large spheres have proportionally less skin.

What is the difference between volume and surface area here?

Volume is 4πr³ over 3 and surface area is 4πr², so one cubes the radius and the other squares it. For a radius of 6 they read 904.78 cubic units and 452.39 square units, which are different quantities despite the similar look.

Related lessons and calculators

• Radius of a cylinder, the solid this one is measured against, and the same rearranging in reverse.
• Volume of a pyramid, the other solid whose formula carries a fraction, and for the same kind of reason.
• Surface area of a cone, the third member of the cylinder, sphere and cone family.
• Circumference of a circle, where the same radius drives everything with no cubing involved.
• Volume of 3D shapes calculator, for running spheres and the rest without the arithmetic.
• the National Council of Teachers of Mathematics, for how volume and scaling are sequenced in the curriculum.
• Volume of a cone, the third solid in that ratio, worked through on its own.
• Surface area of a sphere, the same solid measured for skin, where the two thirds rule turns up again.
• Volume of a cylinder, the snug container a sphere fills exactly two thirds of.