About the Capsule Calculator
A capsule — also called a stadium of revolution or spherocylinder — is a cylinder capped at both ends by hemispheres of the same radius. The shape turns up in pressure vessels and propane tanks, pill and gel-cap shapes, rounded storage tanks, bacteria in biology and collision shapes in game physics. This capsule calculator returns its volume and surface area, together with the split between the cylindrical body and the rounded ends.
Enter the radius (or diameter) and either the length of the straight cylindrical section or the overall end-to-end length; the calculator converts between the two. If the straight section is zero the capsule is simply a sphere.
Use one length unit throughout; volume comes out in cubic units and area in square units. In centimetres, cm³ equals millilitres, and the notes convert the result to litres and US gallons for tank sizing.
With the default inputs, the volume is 108.9085 cu units. Change any value above to recalculate instantly.
How to use the capsule calculator
- 1Choose whether you know the radius or the diameter, and enter it.
- 2Choose whether your length is the straight section or the overall end-to-end length.
- 3Enter the length in the same unit.
- 4Read the volume and surface area, and the split between the body and the ends.
Formula and method
A capsule is a cylinder of radius r and straight length a with a hemisphere on each end. The two hemispheres together form one complete sphere, so the volume is the cylinder volume πr²a plus the sphere volume (4/3)πr³. The surface area is the cylinder’s side 2πra plus the sphere’s surface 4πr², which factors to 2πr(2r + a). The flat cylinder ends are not counted because they are covered by the hemispheres.
If you enter the overall length, the straight section is found by subtracting the two radii: a = overall − 2r. When a is 0 the shape reduces to a sphere.
- r
- Radius of the cylinder and the hemispherical ends
- a
- Length of the straight cylindrical section
- V
- Volume
- A
- Surface area
Worked examples
Radius 2 with a 6-unit straight section
The cylinder holds π × 4 × 6 ≈ 75.40 and the two ends form a sphere of 4/3 × π × 8 ≈ 33.51, for a total of about 108.91. The surface is 2π × 2 × (4 + 6) ≈ 125.66.
Tank 3 ft in diameter and 10 ft long overall
r = 1.5 ft and the straight section is 10 − 3 = 7 ft. V = π × 2.25 × (2 + 7) ≈ 63.62 ft³ — about 476 US gallons at 7.48 gal per cubic foot.
Zero straight section: a sphere of radius 5
With no cylinder, the capsule is a sphere: V = 4/3 × π × 125 ≈ 523.60 and A = 4π × 25 ≈ 314.16.
Frequently asked questions
What is the formula for the volume of a capsule?+
V = πr²a + (4/3)πr³, where r is the radius and a the length of the straight section. This equals a cylinder plus one whole sphere made from the two ends.
What is the surface area of a capsule?+
A = 2πr(2r + a): the cylinder side 2πra plus a sphere’s 4πr². For r = 2 and a = 6 that is 2π × 2 × 10 ≈ 125.66.
How do I use the overall length instead of the straight length?+
Subtract the diameter: the straight section is the overall length minus 2r. A 10 ft long tank with a 3 ft diameter has a 7 ft straight section.
How many gallons does a capsule tank hold?+
Measure in inches and divide the cubic-inch volume by 231 for US gallons, or measure in feet and multiply cubic feet by about 7.48. For partially filled horizontal tanks use the tank volume calculator.
Is a pill capsule the same shape?+
Pharmaceutical capsules are close to this shape, but real shells have overlapping caps and slightly domed rather than perfect hemispherical ends, so the geometric volume is an approximation.