What is a capsule?
A capsule is a three-dimensional shape made of a right circular cylinder of radius r and length (side length) a, with a hemisphere of the same radius r attached to each flat end. Because the two hemispheres together form one complete sphere, a capsule is simply a cylinder plus a sphere. When the side length a equals zero, the capsule reduces to a perfect sphere. This shape is common in pharmaceutical pills, pressure vessels, and storage tanks.
How to use this calculator
Pick a calculation mode that matches the two values you already know — for example "Given r, a" to find volume, surface area and circumference, or "Given r, V" to solve for the side length. Enter your two known values, optionally choose a unit label and number of significant figures, then read off all five properties: radius r, side length a, volume V, surface area S, and circumference C. The unit label is descriptive only — the tool computes in whatever units you type, tagging areas with the square and volumes with the cube of that unit.
The formulas explained
The volume adds the cylinder's volume to the sphere formed by the caps: $$V = \pi r^2 a + \tfrac{4}{3}\pi r^3$$. The surface area adds the cylinder's lateral surface to the full sphere's surface: $$S = 2\pi r a + 4\pi r^2$$. The circumference is that of the circular cross-section: $$C = 2\pi r$$. To solve for a from a known volume, rearrange to \(a = (V - \tfrac{4}{3}\pi r^3) / (\pi r^2)\); from a known surface area, \(a = (S - 4\pi r^2) / (2\pi r)\). To find r from circumference, \(r = C / (2\pi)\).
Worked example
For \(r = 2\) cm and \(a = 5\) cm with \(\pi = 3.14159265359\): $$C = 2\pi(2) = 12.5664 \text{ cm}$$ $$S = 2\pi(2)(5) + 4\pi(2^2) = 36\pi = 113.097 \text{ cm}^2$$ $$V = \pi(2^2)(5) + \tfrac{4}{3}\pi(2^3) = 30.6667\pi = 96.3424 \text{ cm}^3$$ (rounded to 6 significant figures).
FAQ
What does "circumference" mean for a capsule? It is the circumference of the circular cross-section, \(C = 2\pi r\).
Why might I get an "invalid input" message? When solving for the side length from a volume or surface area, the given value must be at least the volume or surface area of a sphere of radius r; otherwise no real cylinder length exists.
Does changing the unit convert the numbers? No — the unit dropdown only labels the output. All inputs must already be in the same length unit.
