Spherical Cap Volume Calculator

A spherical cap is the portion of a sphere cut off by a flat plane. Take one horizontal cut through a watermelon and either piece is a spherical cap, the small one and the large one both.

V = (π × h² × (3R − h)) / 3

Where:

• R = full sphere radius
• h = cap height (the height of the dome from the cutting plane to the top of the sphere)

When h = R, the cap is exactly a hemisphere. When h = 2R, the “cap” is the whole sphere. When h is small, the cap is a thin disc-like sliver.

Worked example: geodesic dome interior volume A backyard greenhouse geodesic dome approximated as a spherical cap: full sphere radius R = 4 m (the design parameter), cap height h = 2.5 m (the actual height to the apex). V = (π × 2.5² × (12 − 2.5)) / 3 = (π × 6.25 × 9.5) / 3 ≈ 62.2 m³.

That is the air volume inside the dome, which is what an HVAC or ventilation calculation starts from.

Worked example: partially-filled spherical water tank A spherical water tank measures 7 ft across, so R = 3.5 ft. Total sphere volume: (4/3)π × 42.875 ≈ 179.6 ft³ ≈ 1,343 gallons.

Watch the naming here. Tanks are sold by a nominal rating that is not the geometric volume, so a tank badged 1,000 gallons and a tank that geometrically holds 1,000 gallons are different objects. Measure the shell and compute, or you will be out by hundreds of gallons. A sphere that genuinely holds 1,000 gallons has R = 3.17 ft, about 6 ft 4 in across.

If the water depth from the bottom is 3 ft, then h = 3 ft. V_water = (π × 9 × (10.5 − 3)) / 3 = (π × 9 × 7.5) / 3 ≈ 70.7 ft³ ≈ 529 gallons.

So 3 ft of water in a 7 ft ball is 43% of the depth but only 39% of the contents. Same trap as a horizontal cylinder: the bottom of the sphere is a narrow bowl, and a foot of depth down there is worth far less than a foot near the equator.

Where spherical caps appear in real measurements:

• Dome buildings and greenhouses. Internal air volume estimation for HVAC.
• Fish-eye camera lens domes. Optical surface design.
• Spherical tank partial fill. Liquid volume in non-full spherical reservoirs (cryogenic, LNG, ammonia).
• Skylight bubbles. Acrylic dome skylights, for material and weight calculations.
• Submersible viewing ports. Hemispherical or capped-spherical view windows.
• Polar ice volume. An ice cap sitting on a spherical Earth, to a first approximation.

Decomposing the formula:

The (3R − h) factor is where the curvature lives. When h is small next to R the cap is a thin lens, 3R − h is near enough 3R, and the whole thing collapses to V ≈ πRh².

That shortcut has a clean reading. A shallow cap’s base radius is close to √(2Rh), so its base circle is about 2πRh in area, and πRh² is exactly half of that base times the height. A thin cap is very nearly a paraboloid, and a paraboloid holds half its bounding cylinder.

The shortcut always runs high, by exactly h / (3R − h). That is 3.4% at h = R/10 and 1.1% at h = R/30, so it is a field estimate rather than a substitute for the real formula.

When h = R (hemisphere), 3R − h = 2R, so V = (π × R² × 2R) / 3 = (2/3)πR³. Matches the hemisphere formula. ✓

When h = 2R (whole sphere), 3R − h = R, so V = (π × 4R² × R) / 3 = (4/3)πR³. Matches the sphere formula. ✓

The relationship between cap height, base radius, and sphere radius:

If a is the base radius of the cap (the radius of the circular cut surface), then: a² + (R − h)² = R² a² = h × (2R − h)

So given the cap height h and sphere radius R, the base radius is: a = √(h × (2R − h))

For our greenhouse example (R = 4, h = 2.5): a = √(2.5 × 5.5) = √13.75 ≈ 3.71 m. That is the floor radius, so the dome covers a circle 7.42 m across.

One last contrast, because it catches people out. The cap’s curved SURFACE is a straight-line function of the height, exactly h / 2R of the whole sphere. The VOLUME is not: it is h²(3R − h) / 4R³. A dome half the height of the ball has half the surface and half the volume, but a dome a quarter of the height has a quarter of the surface and only about 16% of the volume.

Sanity check:

• h = 0: V = 0 (no cap). ✓
• h = R: V = (2/3)πR³ (hemisphere). ✓
• h = 2R: V = (4/3)πR³ (full sphere). ✓