Spherical Cap Surface Area Calculator

A spherical cap has two surfaces, the curved dome and the flat base circle it stands on. This calculator returns both, plus the total.

Curved surface: A_curved = 2 × π × R × h

Flat base: A_base = π × a² where a = √(h × (2R − h)) is the base radius

Total: A_total = 2πRh + πa²

Where R is the full sphere radius and h is the cap height.

The surprise is in the curved term. Slice a sphere with two parallel cuts h apart and you get 2πRh of curved surface no matter WHERE along the axis you cut. A slab taken off the top of the ball and a slab of the same thickness taken from the fat middle have identical curved area, even though one has a tiny base circle and the other has a huge one. This is Archimedes’ Hat-Box Theorem, and it is why the formula never asks you for the base radius.

The consequence for a cap is worth writing down on its own: the curved surface is h / 2R of the whole sphere. Cut a third of the way down the diameter and you have exactly a third of the surface. Volume does nothing of the sort, which the spherical cap volume page goes into.

Worked example: dome paint coverage A geodesic greenhouse dome: R = 4 m, h = 2.5 m. Curved (dome) surface: 2π × 4 × 2.5 = 20π ≈ 62.83 m². Base radius: √(2.5 × 5.5) = √13.75 ≈ 3.71 m. Base area: π × 13.75 ≈ 43.20 m².

If you are painting the OUTSIDE of the dome only, with no floor, that is 62.83 m². Read the coverage figure off the tin you actually buy, since it swings a long way between products, and double it for two coats.

Worked example: fish-eye dome skylight A pop-up acrylic skylight bubble: R = 30 cm (sphere radius), h = 20 cm (dome height). Curved surface: 2π × 30 × 20 = 1,200π ≈ 3,770 cm² = 0.377 m².

That’s the acrylic sheet area needed to thermoform the bubble. Add 30% for trim waste and edge mounting: ~0.5 m² per bubble. Sheet acrylic is sold by the square meter; one 4×8 ft sheet (~3 m²) yields about 6 bubbles.

Where spherical cap surface matters:

• Dome paint and coating coverage. Geodesic domes, planetarium domes, observatory housings.
• Acrylic skylight and bubble fabrication. Thermoformed dome sheet area.
• Submarine and submersible viewing ports. Acrylic or fused-silica dome thickness × surface area gives material weight.
• Camera fish-eye lens dome surface. Anti-reflective coating coverage.
• Half-spherical tank end cap painting. When h = R (a hemisphere) on the end of a cylindrical tank.
• Contact lens and watch-crystal blanks. A shallow cap on a very large R, where h is under a millimetre.

Archimedes’ Hat-Box Theorem in detail:

Take a sphere of radius R and circumscribe it with a cylinder of radius R and height 2R. Now cut both shapes with two parallel horizontal planes spaced h apart.

Archimedes proved that the curved surface area of the spherical slice equals the curved surface area of the cylindrical slice. Both equal 2πRh.

Archimedes rated this above everything else he did, and asked for the sphere-in-a-cylinder to be carved on his tomb rather than any of his machines. Cicero went looking for that tomb as quaestor in Sicily in 75 BC, 137 years after Archimedes was killed, and found it overgrown in a thicket outside Syracuse with the carving still legible.

Hemisphere as a special case:

When h = R the cap is a hemisphere and the curved surface is 2πR², exactly half the full sphere’s 4πR². The h / 2R rule predicts that with no work at all: h = R means h / 2R = ½.

Sanity check:

• h = 0: A_curved = 0, A_base = 0. ✓
• h = R (hemisphere): A_curved = 2πR², A_base = πR². Total = 3πR². ✓
• h = 2R (full sphere): A_curved = 4πR², A_base = 0. Total = 4πR². ✓