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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.

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))

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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