Sphere Calculator

What this sphere calculator does

A sphere is the set of all points at the same distance from a centre, and that single distance, the radius, is enough to describe it completely. Every other measurement follows from it. That is why this calculator asks for just one quantity: the radius, the diameter, the surface area or the volume. Pick which one you have under Solve using, type the value, and the other three are calculated for you.

The input has its own unit, and so does the output. You can enter a radius in millimeters and read the results in meters, or feed in a volume in cubic feet and get a radius in inches. Conversion happens before any geometry, so mixed units never cause trouble.

Sphere formulas

With radius r and diameter d = 2r, the three results are linked by these relations, where π ≈ 3.14159.

volume = 4⁄3 × π × r³ = π × d³ ÷ 6

surface area = 4 × π × r² = π × d²

diameter = 2 × radius

Volume is measured in cubic units (cm³, ft³) because it has three dimensions, and surface area in square units (cm², ft²) because it has two. The calculator labels each answer with the matching unit so you do not have to track the exponent yourself.

Finding the radius from area or volume

When the known quantity is the surface area or the volume, the radius comes from undoing the formulas above. The area formula needs a square root, the volume formula a cube root.

r = √( A ÷ (4π) )radius from surface area A

r = ∛( 3V ÷ (4π) )radius from volume V

Once r is known, the remaining quantities are straightforward. There is also a handy shortcut linking the two main results: volume = surface area × radius ÷ 3. It is a quick way to sanity-check an answer, and it works because 4πr² × r ÷ 3 simplifies to 4⁄3πr³.

How to use the calculator

1. Open Solve using and choose Radius, Diameter, Surface area or Volume, whichever you know.
2. Type the number in the field that appears and select its unit. Area and volume inputs use square and cubic units.
3. Under Output units choose the length unit for the results; areas and volumes follow it automatically.
4. Press Calculate. Answers are shown with up to four decimal places, and whole numbers appear without any.

The answer list always leaves out the quantity you entered and shows the other three. If you start from the radius you get volume, area and diameter; starting from the volume gives area, radius and diameter.

Worked examples

From the radius. A decorative glass float has a radius of 7.5 cm. Volume = 4⁄3 × π × 7.5³ = 1,767.1459 cm³, which is about 1.77 liters. Surface area = 4 × π × 7.5² = 706.8583 cm², and the diameter is 2 × 7.5 = 15 cm.

From the diameter. A ball measures 22 cm across. Its volume is π × 22³ ÷ 6 = 5,575.2798 cm³ (about 5.58 liters) and its surface area is π × 22² = 1,520.5308 cm². The radius is half the diameter, 11 cm.

From the volume. A spherical tank must hold exactly 1,000 cm³. Then r = ∛(3 × 1000 ÷ 4π) = ∛238.7324 ≈ 6.2035 cm, the diameter is 12.4070 cm, and the surface area is 4π × 6.2035² ≈ 483.5976 cm². Check with the shortcut: 483.5976 × 6.2035 ÷ 3 ≈ 1,000.

How size changes the results

Doubling the radius multiplies the surface area by 4 and the volume by 8. Bigger spheres therefore hold disproportionately more than they cost to cover. The table shows the effect for a few radii in centimeters.

Common mistakes to avoid

• Mixing up radius and diameter. Halving or doubling by mistake changes the volume by a factor of 8. Check which one your measurement is, then pick it under Solve using.
• Reading cubic units as square units. A volume of 500 cm³ and an area of 500 cm² describe very different things; the input unit selector must match the quantity.
• Rounding π too early. Using 3.14 instead of the full value is fine for rough work but shows up in the third or fourth significant digit.
• Forgetting the output unit. The same sphere is 4,188.79 cm³ and 4.18879 liters; make sure the unit in the answer is the one you need.

For the volume of a sphere on its own, see the volume of a sphere calculator. To compare with other solids, try the volume calculator or the surface area calculator.