Sphere Volume & Surface Area Calculator
A sphere is fixed by any one of radius, diameter, surface area or volume. Pick which one you know, type the value, and the tool uses Archimedes' formulas V = (4/3)πr³ and S = 4πr² to recover the other three instantly — plus the "great circle" (a slice through the centre) so you can answer follow-ups like "what is the cross-section of this ball?" without re-deriving the algebra.
Enter a positive number.
Radius r
10
Given
Diameter d
20
Computed
Surface area S
1256.6371
Computed
Volume V
4188.7902
Computed
Great-circle circumference
62.8319
Circumference of a slice through the centre (the equator), C = 2πr.
Great-circle area
314.1593
Area of that same centre slice, A = πr².
Formula
d = 2r ; S = 4πr² ; V = (4/3)πr³
Pick any unit (cm, m, in, ft, km). Radius and diameter share one linear unit; surface area uses that unit squared; volume uses that unit cubed. 1 litre = 1000 cubic centimetres.
Formula
d = 2r ; S = 4πr² ; V = (4/3)πr³ — invert: r = d/2 = √(S/(4π)) = ∛(3V/(4π))
- · Radius and diameter share one linear unit (cm, m, in, ft, km — your choice); surface area uses that unit squared and volume uses it cubed.
- · Unit conversions: 1 litre = 1000 cm³ = 0.001 m³; 1 US gallon ≈ 3.785 L; 1 ft³ ≈ 28.317 L.
- · Sphere surface = 4 × great-circle area (Archimedes), and sphere volume = (2/3) × volume of the smallest enclosing cylinder — the result Archimedes was proudest of.
- · A "great circle" is any slice through the centre; its radius equals the sphere's radius. Earth's equator and meridians are great circles, so the great-circle circumference is the longest possible circle you can draw on the surface.
- · Only positive numbers are accepted; zero or negative values show an error. All arithmetic uses double precision, with π = Math.PI ≈ 3.141592653589793.
- · Formulas: Archimedes, On the Sphere and Cylinder, Book I, Propositions 33 and 34; CRC Standard Mathematical Tables, 31st ed., §4.10.
Frequently asked
Where does the (4/3)πr³ formula for sphere volume come from?
Archimedes proved it first by the method of exhaustion: place the sphere inside the smallest cylinder that contains it (height 2r, base radius r), and the sphere takes up exactly 2⁄3 of that cylinder. Cylinder volume = πr² × 2r = 2πr³, so the sphere = (2/3) × 2πr³ = (4/3)πr³. The modern derivation is a one-line integral: V = ∫₋ᵣʳ π(r² − x²) dx = (4/3)πr³.
I only know the volume — can I recover the radius?
Yes. Inverting V = (4/3)πr³ gives r = ∛(3V / (4π)). Pick "Known = Volume" in the tool and it shows radius, diameter and surface area in one go. Example: a 1-litre ball (1000 cm³) has radius ∛(3000 / (4π)) ≈ 6.20 cm, diameter ≈ 12.41 cm.
Is "great-circle circumference" the same as "the sphere's circumference"?
In everyday speech "the sphere's circumference" means the great-circle circumference — the longest loop you can wrap around it. Every great circle has the same length, 2πr. Earth's equator is roughly 40,075 km, exactly what you get from 2πr with the mean radius ≈ 6,371 km. Strictly the sphere itself has no edge, so the quantity only makes sense on a circle that passes through the centre.
One ball has twice the radius of another — how much bigger is its volume?
Volume scales with the cube of the radius (V ∝ r³), so doubling the radius gives 2³ = 8 times the volume. Surface area scales with the square (S ∝ r²), so it grows fourfold. Concrete example: an NBA size-7 basketball (d ≈ 24 cm) has V ≈ 7.24 L, a tennis ball (d ≈ 6.7 cm) has V ≈ 0.16 L — about a 45× ratio overall.
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