Sphere Surface Area

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.