Sphere Calculator
Volume and surface area of a sphere from radius or diameter — or solve the radius back from a known volume or area.
A sphere of radius r has volume V = 4⁄3·π·r³ and surface area SA = 4·π·r², where the diameter is 2r. For r = 5, that gives V = 4⁄3·π·125 ≈ 523.6 and SA = 4·π·25 ≈ 314.16. Enter any one measure and the tool returns the other three.
What a sphere calculator does
A sphere is fixed by a single number — its radius r (or, equivalently, its diameter d = 2r). From that one value every other measure follows: the surface area SA = 4πr² and the volume V = 4⁄3πr³. This tool also runs the two formulas backwards, so you can start from a known volume or surface area and recover the radius.
volume scales with r³, surface area with r²; diameter d = 2r
Worked example
Take a sphere of radius r = 5:
- 1 Note the radius. r = 5, so the diameter is d = 2r = 10.
- 2 Cube the radius for the volume. V = 4⁄3·π·r³ = 4⁄3·π·5³ = 4⁄3·π·125 ≈ 523.60.
- 3 Square the radius for the surface area. SA = 4·π·r² = 4·π·5² = 4·π·25 ≈ 314.16.
- 4 To reverse it, solve for r. From volume: r = ∛(3V ÷ 4π). From area: r = √(SA ÷ 4π). Both return r = 5.
Sphere volume and surface area by radius
Volume grows with the cube of the radius; surface area with the square. At r = 3 the two happen to coincide (both 36π ≈ 113.10).
| Radius r | Diameter 2r | Volume 4⁄3πr³ | Surface area 4πr² |
|---|---|---|---|
| 1 | 2 | 4.19 | 12.57 |
| 2 | 4 | 33.51 | 50.27 |
| 3 | 6 | 113.10 | 113.10 |
| 5 | 10 | 523.60 | 314.16 |
| 10 | 20 | 4188.79 | 1256.64 |
Reading the results
Why the 4⁄3? The volume of a sphere is exactly two-thirds of the cylinder that just encloses it — a result Archimedes proved. Working it out with calculus (integrating circular cross-sections, or the shell method) produces the constant 4⁄3, so V = 4⁄3πr³ rather than the πr² of a flat disc.
Sphere vs circle. A circle is two-dimensional: it has area πr² and circumference 2πr. A sphere is the three-dimensional surface at distance r from a point, so it instead has a surface area (4πr²) and a volume (4⁄3πr³). Notice the sphere’s surface area is exactly four times the circle’s area — a neat link between the two shapes.
Units. Whatever unit you use for r carries through: length for the diameter, square units for surface area, cubic units for volume. Because volume depends on r³, doubling the radius multiplies the volume by 8 while the surface area only quadruples.