Hemisphere Volume Calculator

A hemisphere is half a sphere, cut along a great circle through the center. The volume is half the sphere’s volume:

V = (2/3) × π × r³

Worked example: half-spherical mixing bowl A 10-inch stainless mixing bowl that is exactly hemispherical. That is rarer than you would think, since most “round” bowls have a flattened or rolled bottom so they sit still on a counter. r = 5 in. V = (2/3) × π × 125 ≈ 261.8 in³ ≈ 1.13 US gallons, or about 4.3 liters.

Reckon on a real bowl holding roughly 80% of the geometric figure, which puts that one near 3.4 liters filled to the rim. You would stop well short of that in practice, since flour goes everywhere the moment you switch the mixer on.

Where hemisphere volumes show up:

• Mixing bowls and salad bowls (when approximated as half-spheres).
• Domed roofs and architecture. A pure hemispherical dome of internal radius r contains (2/3)πr³ of interior volume.
• Half-buried propane tanks. Some pressure vessels have hemispherical end caps. The end cap volume is (2/3)πr³.
• Geodesic domes. Used in greenhouses and exhibition halls, and close enough to a hemisphere for a first pass at the enclosed air volume.
• Planetarium and observatory domes. The interior volume drives the heating and cooling load, which is the number the building engineer wants.

Hemisphere vs. half-spherical “cap”:

A hemisphere is exactly half, so the cap height equals the radius. A “spherical cap” is the more general shape, where the cap height can be anything from 0 (a thin slice) up to 2r (the full sphere). At h = r the spherical cap collapses into the hemisphere. Below that you get a flatter dome slice, and the volume drops away faster than the height does.

Comparing hemisphere to other half-shapes:

• Half a cube (cut horizontally): V = ½ × s³.
• Half a cylinder (cut along the axis): V = ½ × π × r² × h.
• Half a sphere (hemisphere): V = (2/3) × π × r³.

So a hemisphere holds 2/3 the volume of a cylinder with the same radius and a height equal to that radius. That is part of the classical Archimedes result: cylinder, sphere and cone stand in the ratio 3:2:1 when all three share a radius and the cone and cylinder both stand 2r tall.

Surface area note (covered in full on its own page): A hemisphere’s surface includes both the curved dome (2πr²) and the flat circular base (πr²), so SA = 3πr². Leave the base off, as you would for a dome roof measured for shingles, and it is 2πr².

Sanity check:

• r = 0: V = 0. ✓
• Hemisphere volume = ½ × sphere volume. ✓ (2/3 = ½ × 4/3.)
• A 1-cm-radius hemisphere has volume 2π/3 ≈ 2.094 cm³.