Sphere Volume Calculator
Type the radius of a sphere — the calculator returns volume, surface area, diameter, circumference and great-circle area, all using exact π.
Volume (V = 4⁄3·π·r³)
523.598776 cubic units
- Diameter (d = 2r)
- 10
- Circumference (C = 2π·r)
- 31.42
- Great-circle area (π·r²)
- 78.54
- Surface area (A = 4π·r²)
- 314.16
A sphere of radius r has volume V = 4⁄3·π·r³ and surface area A = 4π·r². Archimedes showed the sphere fills exactly two-thirds of its circumscribed cylinder. The great-circle cross-section through the centre has area π·r² and circumference 2π·r.
How to use this calculator
Type the radius r of the sphere in any linear unit you like — centimetres, metres, inches or feet. The calculator returns the volume in cubic units of that unit, alongside the diameter, circumference of a great circle through the centre, great-circle area, and the total surface area. Linear outputs use the same unit, areas use squared units, the volume uses cubed units. π is taken at full machine precision, so the only rounding is at display.
How the calculation works
A sphere is the set of all points in space at distance r from a single centre. From r alone every other measurement follows: the diameter is d = 2r, any great circle through the centre has circumference C = 2π·r and encloses an area π·r². The surface area is A = 4π·r² — exactly four times the area of a great circle, a result Archimedes proved in the 3rd century BC. The volume is V = 4⁄3·π·r³, which Archimedes also showed equals two-thirds the volume of the smallest cylinder that fits around the sphere (radius r, height 2r). Both the 4 in the surface area and the 4⁄3 in the volume can be derived by integration in spherical coordinates.
Worked example
Take r = 3. Diameter = 2·3 = 6. Circumference = 2π·3 = 6π ≈ 18.849556. Great-circle area = π·3² = 9π ≈ 28.274334. Surface area = 4π·3² = 36π ≈ 113.097336. Volume = 4⁄3·π·3³ = 36π ≈ 113.097336 cubic units. (For r = 3 the numerical values of surface area and volume coincide — that only happens at r = 3.) If your radius is in centimetres, the volume is in cm³ (millilitres), so a 3 cm sphere holds about 113 mL; in metres it is in m³; in inches it is in cubic inches (1 in³ ≈ 16.387 mL).
Frequently asked questions
What is the formula for the volume of a sphere?
V = 4⁄3·π·r³, where r is the radius — the distance from the centre of the sphere to its surface. The 4⁄3 factor is exact, not an approximation; it falls out of integrating the area of circular cross-sections π·(r² − z²) along the axis from −r to r. Archimedes proved it geometrically in the 3rd century BC using the method of exhaustion.
What is the surface area of a sphere?
A = 4π·r² — exactly four times the area of a great circle through the centre. A neat consequence: the surface area equals the lateral surface of the smallest cylinder that fits around the sphere (radius r, height 2r), which is 2π·r·2r = 4π·r². This is Archimedes' hat-box theorem and is the reason a sphere and its circumscribed cylinder share the same surface area.
How is volume different from surface area?
Volume measures how much three-dimensional space the sphere occupies (cubic units — cm³, m³, in³). Surface area measures the two-dimensional skin around it (squared units — cm², m², in²). For a sphere both depend only on the radius, but they scale differently: doubling r multiplies surface area by 4 and volume by 8. The two only happen to take the same numerical value when r = 3 — pure coincidence of the units, not a deep result.
How do I convert the volume to litres or gallons?
If your radius is in centimetres, the volume comes out in cubic centimetres, and 1 cm³ = 1 millilitre, so divide by 1000 to get litres. In metres, the volume is in m³ and 1 m³ = 1000 litres. For US gallons, 1 US gallon ≈ 3.78541 litres; for UK gallons, ≈ 4.54609 litres. A football (soccer ball) of radius ~11 cm holds about 5.6 litres of air.
How does a sphere relate to its circumscribed cylinder?
A sphere of radius r fits snugly inside a cylinder of radius r and height 2r. The cylinder has volume π·r²·2r = 2π·r³ and the sphere has volume 4⁄3·π·r³, so the sphere is exactly two-thirds of the cylinder. Their lateral surface areas are both 4π·r². Archimedes regarded this 2:3 ratio as his greatest discovery and asked for a sphere-in-cylinder to be carved on his tombstone.
What if I know the diameter instead of the radius?
Divide the diameter by 2 to get the radius, then enter it. Alternatively, the formulas in terms of diameter d are: V = π·d³⁄6, A = π·d², C = π·d. A ball quoted as "10 cm across" has d = 10, so r = 5, volume = 4⁄3·π·5³ ≈ 523.6 cm³ (≈ 0.52 litres).