What Is a Capsule?
A capsule (also called a stadium of revolution or discorectangle of revolution) is a three-dimensional shape made of a cylinder with two hemispherical caps on each end. It is the shape of a pharmaceutical pill, a propane tank, or a pressure vessel. This calculator computes both the volume and the surface area of a capsule from two measurements: the radius r of the hemispheres (and cylinder) and the length a of the straight cylindrical section.
How to Use the Calculator
Enter the radius r and the cylinder length a in the same units (e.g. millimetres, centimetres or inches). The calculator returns the volume in cubic units and the surface area in square units. Note that a is the length of only the cylindrical middle — the total tip-to-tip length of the capsule equals \(a + 2r\).
The Formula Explained
The capsule is the sum of a cylinder and two hemispheres (which together form one full sphere):
Volume:
$$V = \pi r^2 a \;(\text{cylinder}) + \tfrac{4}{3}\pi r^3 \;(\text{sphere}) = \pi r^2 \left(\tfrac{4}{3}r + a\right).$$Surface area:
$$S = 2\pi r a \;(\text{curved cylinder wall}) + 4\pi r^2 \;(\text{sphere}) = 2\pi r (2r + a).$$The flat circular ends are not counted because they are covered by the hemispherical caps.
Worked Example
Suppose \(r = 5\) and \(a = 10\). Volume:
$$V = \pi \cdot 25 \cdot \left(\tfrac{20}{3} + 10\right) = \pi \cdot 25 \cdot 16.6667 \approx 1308.997 \text{ cubic units}.$$Surface:
$$S = 2\pi \cdot 5 \cdot (10 + 10) = 2\pi \cdot 5 \cdot 20 = 200\pi \approx 628.319 \text{ square units}.$$FAQ
Is a the total length? No. a is only the cylindrical part. Total length = \(a + 2r\).
What if a = 0? The capsule becomes a sphere, and the formulas reduce to \(V = \tfrac{4}{3}\pi r^3\) and \(S = 4\pi r^2\).
What units does it use? Any consistent unit — results are simply in cubic and square versions of your input unit.
