Hemisphere Surface Area Calculator
Total and curved surface area of a hemisphere (half sphere) from radius or diameter, including hollow hemispheres.
A hemisphere’s surface area — the shape of a dome roof, a half-ball, or a rounded tank end — splits into two parts: the flat circular base and the curved cap rising above it. This calculator solves for both, plus the total surface area and the volume, all from a single measurement. Enter whichever value you already know — radius, diameter, base area, curved (cap) area, total surface area, or volume — and the rest solve automatically.
This reverse-solving flexibility matters because a hemisphere isn’t always specified by its radius directly — a dome’s rated interior volume or a spec sheet’s total-coverage area are often what you start with instead.
Hemisphere Surface Area Formula
Base Area (the Flat Circle)
Base Area = πr²
Curved (Cap) Area
Cap Area = 2πr²
The curved cap is exactly twice the flat base’s area — a consequence of Archimedes’ classic result that a hemisphere’s curved surface equals the surface of the cylinder that tightly encloses it.
Total Surface Area
Total Surface Area = Base Area + Cap Area = 3πr²
Volume
Volume = (2/3)πr³
By Radius vs. By Diameter
Since many real-world measurements (a dome’s span, a tank’s opening width) come as a diameter rather than a radius, you can enter either directly — the calculator divides by 2 first when needed, then applies the same formulas.
Worked Example: By Radius
A hemisphere with a 6-ft radius: Base Area = π × 6² ≈ 113.10 square feet, Cap Area = 2 × π × 6² ≈ 226.19 square feet, Total SA = 3 × π × 6² ≈ 339.29 square feet, Volume = (2/3) × π × 6³ ≈ 452.39 cubic feet.
Worked Example: By Diameter
A hemisphere with a 16-ft diameter: Radius = 16 ÷ 2 = 8 ft, Base Area = π × 8² ≈ 201.06 square feet, Cap Area = 2 × π × 8² ≈ 402.12 square feet, Total SA = 3 × π × 8² ≈ 603.19 square feet, Volume = (2/3) × π × 8³ ≈ 1072.33 cubic feet.
Solving from Volume (Reverse Lookup)
If you only know a dome’s rated interior volume, the calculator finds the radius first and then reports every area measurement from it.
Worked Example: Starting from Volume
A hemisphere with a volume of 1,526.81 cubic feet: Radius = ∛(3 × 1526.81 ÷ 2π) = 9 ft, Base Area = π × 9² ≈ 254.47 square feet, Cap Area = 2 × π × 9² ≈ 508.94 square feet, Total SA = 3 × π × 9² ≈ 763.41 square feet.
Hemisphere vs. Dome — Are They the Same Shape?
A spherical dome is simply a hemisphere used as a roof or ceiling, so the same formulas apply directly. An elliptical dome, however, is a different shape — flattened or stretched rather than perfectly spherical — and its surface area requires a more complex, non-closed-form calculation that this calculator doesn’t cover. If your dome is described as elliptical rather than spherical, these formulas won’t give an accurate result.
Surface Area vs. Volume
These answer different practical questions. Surface area — specifically the total or the cap area — is what you need for a material estimate that covers the dome’s exterior, like roofing or cladding. Volume is what you need for interior capacity, like the air space under a dome roof or the liquid a rounded tank end can hold. Since both come from the same radius, this calculator always reports them together.
Frequently Asked Questions
What’s the difference between curved (cap) area and total surface area?
Curved (cap) area (2πr²) covers only the rounded dome surface, not the flat base. Total surface area adds the flat base circle on top of that: Total SA = Base Area (πr²) + Cap Area (2πr²) = 3πr².
Is a dome’s surface area the same as a hemisphere’s?
Yes, for a spherical dome — the same 3πr² total surface area formula applies. An elliptical dome is a different, flattened or stretched shape that needs its own non-closed-form calculation and isn’t covered here.
How do I find a hemisphere’s surface area from its diameter?
Enter the diameter directly. The calculator divides it by 2 to get the radius, then reports the base area, cap area, total surface area, and volume together.