Spherical Cap Surface Area Calculator

A spherical cap has two surfaces — the curved (dome) surface and the flat (base) circle. 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.

A surprising fact: the curved surface 2πRh is INDEPENDENT of the cap base radius — it depends only on R and h. This is Archimedes’ “Hat-Box Theorem”: any horizontal slice of a sphere of height h has the same curved surface area as a cylinder slice of the same height.

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’re painting the OUTSIDE of the dome only (no floor): 62.83 m². At 8 m²/L per coat, two coats use 16 L of paint. Buy 20 L to be safe.

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 (hemisphere) on the end of a cylindrical tank.

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.

This is one of the most beautiful results in classical geometry. It’s the reason Archimedes asked for a sphere inscribed in a cylinder to be carved on his tomb. (Cicero found and described the tomb 137 years after Archimedes’ death — the carving was still visible.)

Hemisphere as a special case:

When h = R (hemisphere), curved surface = 2πR². That’s exactly half of the full sphere’s 4πR² surface — as expected.

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. ✓