Hemisphere Surface Area Calculator

A hemisphere has two distinct surfaces: the curved dome on top, and the flat circular base underneath.

SA (total) = 3 × π × r²

This breaks down as:

• Curved (dome) surface: 2πr², exactly half of a full sphere’s 4πr².
• Flat (base) circle: πr², the great-circle disc that cut the sphere in half.
• Sum: 2πr² + πr² = 3πr².

Worked example: painting a domed roof, inside and out A small church has a hemispherical dome with interior radius 4 m. Exterior dome surface (rough estimate, ignoring the shell thickness): 2π × 16 = 100.5 m². Interior dome surface: the same again, 100.5 m². Both surfaces, two coats each: 4 × 100.5 = 402 m² of paint coverage.

Industrial exterior paint covers about 8 m² per liter per coat, so 402 / 8 is roughly 50 L for the job. Add 20% for waste and touch-ups and buy 60 L.

Worked example: coating the inside of a mixing bowl A hemispherical stainless mixing bowl, 24 cm interior diameter (r = 12 cm). Interior curved surface (food-grade coating): 2π × 144 = 904.8 cm². Base flat disc (also coated): π × 144 = 452.4 cm². Total inside surface: 1,357 cm² = 0.136 m² to coat.

Where hemisphere surface area matters in practice:

• Dome paint coverage. Interior or exterior, the curved-dome formula is 2πr² with no base. Add the base only if you are coating the floor too.
• Bowl interior finishing. Food-grade epoxy, ceramic glaze, non-stick coating.
• Hemispherical pressure vessel end caps. Some tank designs have hemi-heads. The curved area is the painting target.
• Geodesic dome covering. Approximated as a hemisphere for fabric or panel coverage.
• Wok or pan coatings. Hemispherical woks: interior coating area is 2πr² plus the small flat area at the bottom.
• Lampshade fabric for half-dome shades. Curved-only surface is 2πr².

The “with base” or “without base” question:

This depends on what you’re calculating:

• Coating, paint, or fabric for a dome BUILDING: usually no base (the floor is concrete or wood, not the curved surface). Use 2πr².
• A solid hemispherical block (paperweight, gear shift knob): total skin area = 3πr².
• A bowl interior (concave dome): curved interior + flat base inside the bowl = 3πr² for the inner surface.

Hemisphere vs. a flat lid of the same diameter:

A flat disc of radius r has 2πr² of skin, counting both faces. A hemisphere of the same radius has 3πr². So swapping a flat lid for a dome costs 50% more paint, sealant or fabric while covering exactly the same πr² of ground.

That is the number that catches people quoting a dome job off a floor plan. The plan shows a circle; the painter is working on half a sphere. Price it from the footprint and you come up a third short.

Sanity check:

• r = 0: SA = 0. ✓
• Curved surface only: 2πr², exactly half of the full sphere’s 4πr². ✓
• Total (with base): 3πr² = curved + base. ✓