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Capsule shape: volume and surface area of a cylinder with round ends

Calculate the volume and surface area of a capsule, a cylinder capped by two hemispheres, like a pill, a propane tank or a pressure vessel.

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Volume

395.841

In cubic units of the lengths entered

Surface area

301.593

Overall length
16.000
Volume of the cylindrical part
282.743
Volume of the two hemispheres (one sphere)
113.097

A cylinder with rounded ends

A capsule, also called a spherocylinder, is a cylinder capped at each end by a hemisphere of the same radius. The shape is everywhere: medicine capsules, propane and LPG tanks, compressed air tanks, submarines, some grain silos, pill-shaped buildings and even certain bacteria such as E. coli.

The formulas

With radius r and length a of the straight cylindrical part:

  • Volume: V = πr²a + (4/3)πr³ = πr²(4r/3 + a)
  • Surface area: A = 2πra + 4πr² = 2πr(2r + a)
  • Overall length: a + 2r

The two hemispheres together make one full sphere, which is why the formulas are simply a cylinder plus a sphere. If you know the overall length L instead, the straight part is a = L − 2r.

Example

A capsule with a radius of 3 and a straight part of 10 (overall length 16):

  • Cylinder: π × 9 × 10 = 282.7
  • Two hemispheres: 4/3 × π × 27 = 113.1
  • Total volume: 395.8 cubic units
  • Surface area: 2π × 3 × (6 + 10) = 301.6 square units

Practical uses

  • Tanks: if the dimensions are in inches, divide the volume by 231 for US gallons; in feet, multiply by 7.48. A tank 2 ft in diameter and 4 ft long overall holds about 10.5 cubic feet, or 78 gallons. Propane tanks are filled to only about 80% of their volume to leave room for expansion.
  • Paint and coatings: the surface area tells you how much coating a tank needs.
  • Pharmacy: gelatin capsule sizes are designed from these shapes; a size 0 capsule holds about 0.68 mL.

Why rounded ends

For a pressurized tank, a hemispherical end has only half the wall stress of the cylindrical part at the same thickness, and it has no sharp corners where cracks can start. That makes the capsule one of the most efficient shapes for holding gas under pressure. Many commercial tanks use slightly flatter, dished ends to save length, so check the real dimensions before relying on the result for a fill volume.

When the straight part shrinks to zero, the capsule becomes a sphere. A stadium is the two-dimensional version: a rectangle with a half circle at each end, whose area is 2ra + πr².

Frequently asked questions

What is a capsule shape?

A cylinder closed at each end by a hemisphere of the same radius, also called a spherocylinder or stadium of revolution. Pills, propane tanks, pressure vessels and some storage silos have this shape.

How are the volume and area calculated?

V = πr²(4r/3 + a) and A = 2πr(2r + a), where r is the radius and a the length of the straight part. The volume is a cylinder πr²a plus one full sphere 4/3πr³ made of the two ends.

Why are tanks built with rounded ends?

A hemispherical end carries pressure with about half the wall stress of the cylinder, so it can be thinner, and it has no corners where stress concentrates. Many tanks use flatter dished ends instead to save space.

How do I convert the volume to liters or gallons?

If the lengths are in centimeters, divide the volume by 1,000 for liters. If they are in inches, divide by 231 for US gallons. In feet, multiply by 7.48 for US gallons.

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Results are estimates provided for information purposes. They do not replace professional advice.

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