Capsule Volume Calculator (Pill Shape)

A capsule (also called a stadium of revolution, or a spherocylinder) is a cylinder with a hemispherical cap on each end. Picture a pharmaceutical pill or a propane tank.

V = π × r² × h + (4/3) × π × r³

Where r is the radius of the cylindrical section (also the radius of each hemispherical cap), and h is the length of the straight cylindrical section only, NOT counting the rounded ends.

The first term is the cylinder volume; the second term is the volume of one full sphere (two hemispheres = one sphere).

Worked example: a 100 lb propane tank A typical 100 lb propane tank has dimensions: r = 6 in, total height ≈ 48 in including the hemispherical end caps. Cylinder height h = 48 − 2 × 6 = 36 in (subtracting the two cap heights). V = π × 36 × 36 + (4/3) × π × 216 = 4,072 + 905 ≈ 4,977 in³ ≈ 21.5 US gallons.

Propane is sold by weight (lbs), but the tank has an 80% fill rule for safety. 100 lb propane at 0.51 g/cm³ liquid density occupies about 88 L = 23.3 gallons. The tank’s geometric capacity is ~21.5 gallons, holds ~17.2 gallons at 80% fill, which is 75 lb of liquid propane, and another 25 lb of vapor in the headspace and lines totals to the marketed “100 lb.”

Where capsule shapes matter:

• Pharmaceutical pills, the “hard capsule” shape that most tablets come in.
• Pressure vessels. Hemispherical end caps handle internal pressure with less wall stress than flat ones, so most LPG tanks, oxygen tanks, and chemical storage vessels use this geometry.
• Submarine and torpedo hull shapes. Hemispherical ends reduce drag.
• Vitamin and supplement capsules. Size 0, the most common adult vitamin, is about r = 3.4 mm with a total length of 21.7 mm, so the cylinder length is ≈ 14.9 mm. Run those through the formula and the geometry gives ≈ 0.71 mL. The published fill capacity for a size 0 shell is 0.68 mL, slightly less, because the two halves of a real capsule telescope into each other rather than meeting edge to edge. Worth remembering whenever you compare a calculated volume against a manufacturer’s spec: they are measuring slightly different things.

The headspace trap for tanks

When calculating capsule tank capacity for liquids, remember that you should fill to about 80-90% to leave thermal expansion space. A 100% geometric fill leaves no room for the liquid to expand on a hot day, and pressure then rises frighteningly fast.

Sanity check:

• h = 0: capsule becomes a full sphere. V = (4/3)πr³. ✓
• r = 0: capsule is a line segment of length h. V = 0. ✓
• For r = h = 1: V = π + 4π/3 = 7π/3 ≈ 7.33 cubic units.

Comparing to a cylinder of the same total length V_cylinder = π × r² × h_total. For h_total = h + 2r (the cylinder same total length as the capsule): V_cylinder = π × r² × (h + 2r) = πr²h + 2πr³. V_capsule = πr²h + (4/3)πr³. Difference = 2πr³ − (4/3)πr³ = (2/3)πr³, the volume lost to rounding the ends.

That (2/3)πr³ is exactly the volume between the cylinder corners and the hemisphere caps, the pieces a rounded end leaves behind. So a cylinder of the same length holds (2/3)πr³ more than the capsule, that being the two corner regions the hemispheres cut away.

The capsule trades that volume for vastly better structural integrity under pressure.