Sphere Calculator

Sphere formulas

Every measure follows from the radius, so the calculator first finds the radius from whatever you know and then works out the rest. The results are rounded to six significant digits and also given in terms of π, which is how textbook answers are often written. Areas are in square units and volumes in cubic units of the length you choose: a radius in inches gives square and cubic inches.

Finding the radius from the volume or area

Run the formulas backwards:

• From the circumference: r = C ÷ 2π. A ball 30 inches around has r = 4.77465 inches.
• From the surface area: r = √(A ÷ 4π). An area of 100 gives r = 2.82095.
• From the volume: r = ∛(3V ÷ 4π). A volume of 1,000 gives r = 6.2035.

The circumference is often the easiest measure to take on a real ball: wrap a tape measure around the widest part. A caliper or two books held against the sides give the diameter. For cube roots and powers by hand, the scientific calculator has both.

Balls, tanks and planets

The capacity line in the result converts the volume to liters and U.S. gallons: 1,000 cubic centimeters is a liter, and 231 cubic inches is a gallon. That is the inside volume of a hollow sphere; for a tank with thick walls, measure the inside diameter. For volumes of gravel, soil or concrete in cubic yards, use the cubic yards calculator; to switch the result between gallons and liters, gallons to liters.

Half a sphere

A hemisphere — a dome, a bowl, the end cap of a tank — has half the volume of the full sphere, 2/3 πr³. Its curved surface is half the sphere’s area, 2πr², and adding the flat circular base, πr², gives 3πr² in total. The result shows all three, so you can price paint for a dome (curved surface only) or material for a closed bowl (both).

Why size matters so much

Volume grows with the cube of the radius and surface area with the square. Double the diameter of a ball and it holds 8 times as much while its skin is only 4 times larger. That is why a large spherical tank is efficient — the most volume for the least wall area of any shape — and why small droplets and bubbles have so much surface for their size. Going the other way, a ball with twice the volume is only 26% wider, because ∛2 = 1.26. To compare two sizes as a percent change, the percentage calculator does the arithmetic.