Hemisphere Surface Area Calculator

A hemisphere has two distinct surfaces — the curved dome on top and the flat circular base on the bottom.

SA (total) = 3 × π × r²

This breaks down as:

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

Worked example — domed roof painting (interior + exterior): A small church has a hemispherical dome with interior radius 4 m. Exterior dome surface (rough estimate, ignoring slight thickness): 2π × 16 = 100.5 m². Interior dome surface: same, 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. You’d need around 50 L of paint for the full job — plus another 20% for waste and touch-ups.

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 (no base). Add base if you’re 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 — fabric or panel coverage.
• Wok or pan coatings. Hemispherical woks: interior coating area is 2πr² + 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. flat disk of the same diameter:

A flat disk of radius r has surface area 2πr² (top + bottom). A hemisphere of radius r has total surface 3πr². The hemisphere has 50% more surface than the flat disk of equivalent footprint. Cooling fins, radiators, and heat sinks exploit this — curved surfaces shed heat better.

Sanity check:

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