Last updated: 2026-05-18

Definition

A sphere is the set of all points in three-dimensional space at a fixed distance — the radius r — from a central point. Its volume is the space enclosed by that surface, and its surface area is the area of the surface itself. Both quantities depend only on the radius, so a single measurement determines every property of the sphere. The same formulas apply to any perfect sphere — a ball, a globe, a spherical tank, a soap bubble, or an idealized planet.

Formulas

For a sphere with radius r:

Property	Formula
Volume	V = (4/3)πr³
Surface area	A = 4πr²
Diameter	d = 2r

Origin of the (4/3)π factor

The factor 4/3 traces back to Archimedes, who proved around 250 BCE that a sphere fits inside a cylinder of the same radius and height 2r, occupying exactly two-thirds of the cylinder's volume. Since the cylinder's volume is πr² × 2r = 2πr³, the sphere's volume is (2/3) × 2πr³ = (4/3)πr³.

A modern calculus derivation reaches the same result by integrating circular cross-sections from −r to r:

V = ∫₋ᵣʳ π(r² − x²) dx = π[r²x − x³/3]₋ᵣʳ = π × (4r³/3) = (4/3)πr³

Origin of the 4πr² surface area

One intuitive route is to differentiate the volume with respect to r. Peeling off an infinitesimally thin shell gives dV/dr = 4πr², which is exactly the surface area. Geometrically, Archimedes showed that the surface area of a sphere equals the lateral area of the enclosing cylinder — the result engraved on his tombstone.

Cubic scaling: doubling the radius

Because volume grows as r³, doubling the radius multiplies the volume by 8, while the surface area, which grows as r², only quadruples:

Radius factor	Volume factor	Surface area factor
×2	×8	×4
×3	×27	×9
×½	×⅛	×¼

A basketball (r ≈ 12 cm) holds about 8× as much air as a tennis ball (r ≈ 6 cm), even though it looks only twice as wide.

Worked examples

Example 1 — Soccer ball

A regulation soccer ball has a circumference of about 68.5 cm, giving a radius of r = 68.5 / (2π) ≈ 10.9 cm.

Volume: V = (4/3) × π × (0.109)³ ≈ 0.00542 m³ ≈ 5.42 litres Surface area: A = 4 × π × (0.109)² ≈ 0.149 m² ≈ 1490 cm²

Example 2 — Storage tank

A spherical water tank with radius 2 m holds:

V = (4/3) × π × 8 ≈ 33.51 m³ ≈ 33,510 litres

Its surface area A = 4π × 4 ≈ 50.27 m² tells the painter how much steel needs coating.

Example 3 — Finding radius from volume

For a sphere that holds exactly 1 m³, the radius is:

r = ∛(3V / (4π)) = ∛(3 / (4π)) ≈ 0.6204 m (about 62 cm)

Frequently Asked Questions (FAQ)

What is the formula for the volume of a sphere?

The volume of a sphere is V = (4/3)πr³, where r is the radius. For example, a sphere with r = 5 cm: r³ = (0.05 m)³ = 0.000125 m³, so V = (4/3) × π × 0.000125 ≈ 0.000524 m³ ≈ 523.6 cm³.

How do you calculate the surface area of a sphere?

The surface area of a sphere is A = 4πr². For a sphere with r = 5 cm, A = 4 × π × 25 = 100π ≈ 314.2 cm².

What happens to the volume if you double the radius?

Because volume scales as r³, doubling the radius multiplies the volume by 2³ = 8. A sphere of r = 10 cm holds eight times as much as one of r = 5 cm. Similarly, tripling the radius gives 3³ = 27 times the volume.

How do you find the radius from the volume of a sphere?

Rearrange V = (4/3)πr³ to get r = ∛(3V / (4π)). For example, if V = 1 m³, then r = ∛(3 / (4π)) ≈ 0.6204 m.