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Sphere Surface Area

S=4πr2S = 4 \pi r^{2}

Worked example: Sphere r = 0.5 m → S = pi m2 — press Try an example to run it live, then adjust anything.

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π=3.141592653589793\pi = 3.141592653589793Pi · exact
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Sphere Surface Area explained

rS

A sphere's surface area is exactly four times the area of its great circle — the disc you get by slicing it through the centre. Archimedes proved this with an elegant argument comparing the sphere to its circumscribing cylinder; in modern language, S = 4πr² is simply the derivative of the volume (4/3)πr³ with respect to r, because growing a sphere adds a thin shell whose volume is surface area times thickness.

A worked example: to paint a hemispherical dome of radius 10 m, the curved surface is half a sphere, S = ½ · 4π(10)² ≈ 628 m² — at 8 m² per litre you would budget roughly 79 litres per coat. Solving for r takes the principal (positive) square root, the only physically meaningful choice since a radius is a positive length. The square-law scaling is why heat loss, drag, and paint budgets all grow four-fold when the radius merely doubles.

Sphere Surface Area formula

S=4πr2S = 4 \pi r^{2}
Where
  • SS= Surface area (m²)
  • rr= Radius (m)

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