Spherical Cap Volume Calculator

A spherical cap is the portion of a sphere cut off by a flat plane. Imagine slicing a watermelon with a single horizontal cut — the smaller piece (or larger piece) is a spherical cap.

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’s the air volume inside the dome — useful for HVAC sizing or ventilation calculations.

Worked example — partially-filled spherical water tank: A 1,000-gallon spherical water tank has radius R = 3.5 ft. Total sphere volume: (4/3)π × 42.875 ≈ 179.6 ft³ ≈ 1,344 gallons. (Manufacturers round to 1,000 gallons because tanks don’t actually fill to overflowing.)

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

The tank is 529 / 1,344 = 39% full when 3 feet of the 7-foot diameter has water in it — same intuition as for horizontal cylindrical tanks: the math isn’t linear with depth.

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 — material and weight calculations.
• Submersible viewing ports. Hemispherical or capped-spherical view windows.
• Earth surface “spherical cap” areas — e.g. polar ice cap as approximation.

Decomposing the formula:

The (3R − h) factor encodes the curvature. When h is small relative to R, the cap is “thin” — almost a flat disc — and 3R − h ≈ 3R. The formula simplifies to approximately V ≈ πRh², which is what you’d get if you approximated the cap as a thin spherical shell.

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’s the floor radius.

Sanity check:

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