Volume of a sphere calculator

The volume of a sphere is V = (4/3)πr³, where r is the radius, the distance from the center to the surface. For a radius of 5 cm: r³ is 125, so V = (4/3) × π × 125 = (500/3)π, about 523.599 cm³. A cubic centimeter is a milliliter, so a ball that size holds about 0.524 liters.

The formula is Archimedes', and he is said to have wanted the figure behind it on his tomb. Put the sphere inside the cylinder that just contains it, radius r and height 2r. The cylinder's volume is πr² × 2r = 2πr³, and the sphere is exactly two-thirds of that: (2/3) × 2πr³ = (4/3)πr³. The proof slices both solids and a cone into thin disks and shows the slices balance; in calculus it is the integral of the disk areas π(r² - x²) from -r to r. The same cylinder gives the surface area: the sphere's surface equals the cylinder's curved side, 2πr × 2r = 4πr², which for this ball is 100π, about 314.16 cm².

Another way to see the size: the sphere fits in a cube of side 10 cm, which holds 1,000 cm³, and fills π/6 of it, about 52.36%. That fraction is the same for every sphere in its box, which is why round fruit packed in a crate leaves so much air.

The mistake with the biggest effect is the diameter. Radius is half the width, and because it is cubed, using the 10 cm diameter in its place gives 4,188.79 cm³, eight times too much. If you measured across the ball, either halve it first or use V = πd³/6. The second mistake is mixing up the two formulas: 4πr² is the surface area, in square units, and (4/3)πr³ is the volume, in cubic units. If the answer is meant to be in cubic centimeters and the formula only squares r, it is the wrong one.

Going backwards, from a volume to a radius, takes a cube root: r = ∛(3V / (4π)). A sphere that holds exactly one liter, 1,000 cm³, has a radius of ∛(750/π), about 6.20 cm. Because volume grows with the cube of the radius, doubling the radius makes a sphere hold eight times as much, while its surface only grows four times.

Half a sphere is a hemisphere, with half the volume, (2/3)πr³, and a total surface of 3πr² once the flat face is counted. A sphere stretched along its axes is an ellipsoid, (4/3)πabc. Both are shapes on this calculator, as is a capsule, which is a cylinder with a hemisphere on each end.