Capsule Calculator: Volume & Surface Area

The three capsule measures

A capsule (also called a pill shape) is a cylinder of radius $r$ and STRAIGHT body length $a$, capped on each end by a hemisphere of the same radius $r$. Given $r$ and $a$, every measure follows directly:

$V = \pi r^2\left(\frac{4}{3}r + a\right) \qquad S = 2\pi r(2r + a) \qquad C = 2\pi r$

Volume is the two hemispherical caps re-assembled into one full sphere, $\frac{4}{3}\pi r^3$, plus the cylindrical body $\pi r^2 a$. Surface area works the same way: the caps' curved surfaces re-assemble into one sphere's surface, $4\pi r^2$, plus the cylinder's lateral (side) area $2\pi r a$.

Worked example: r = 3, a = 10

$V = \pi(9)(4 + 10) = 126\pi \approx 395.8407 \qquad S = 2\pi(3)(6 + 10) = 96\pi \approx 301.5929 \qquad C = 2\pi(3) \approx 18.8496$

Because $r = 3$ and $a = 10$ are both exact whole numbers here, volume and surface area both have an EXACT symbolic form ($126\pi$, $96\pi$) in addition to the decimal approximation.

Worked example: a = 0 (the sphere boundary case)

$V = \pi(9)\left(4 + 0\right) = 36\pi \approx 113.0973 \qquad S = 2\pi(3)(6 + 0) = 36\pi \approx 113.0973$

Setting the body length to $a = 0$ collapses the cylindrical middle section entirely, leaving just the two hemispherical caps meeting at a single seam — a bare sphere of radius $r = 3$. Both the volume and surface area formulas reduce exactly to the sphere's own $\frac{4}{3}\pi r^3$ and $4\pi r^2$, and both come out to the same coefficient ($36\pi$) here only because $r = 3$ makes $\frac{4}{3}r = 4 = 2r - 2$; that's a coincidence of this particular radius, not a general rule.

Common mistakes

• Mistaking a for the capsule's total length. a is the straight body only — the true overall length is $a + 2r$, one radius longer on each end for the domes.
• Dropping either the caps or the body. Volume is ALWAYS sphere-caps ($\frac{4}{3}\pi r^3$) plus cylinder-body ($\pi r^2 a$) — never one term alone.
• Using the wrong volume coefficient for the caps. The two caps together contribute $\frac{4}{3}\pi r^3$ (one full sphere), not $\frac{2}{3}\pi r^3$ (that's just ONE hemisphere).

Where this shows up

• Capsule-shaped pills: the namesake shape — a cylindrical body with two rounded ends, sized to be easy to swallow while holding more medicine than a plain sphere of the same width.
• Tanks and pressure vessels: propane tanks, compressed-gas cylinders, and some submarine hulls use rounded (hemispherical or capsule) ends instead of flat caps, since a dome resists internal pressure far better than a flat disk of the same radius.
• Building block for other solids: a capsule is exactly a sphere (the two caps) plus a cylinder (the body), and it's the 3D analog of the 2D stadium shape.

A capsule is defined by just two measures, radius $r$ and STRAIGHT body length $a$ — every formula here is the sphere-cap terms ($\frac{4}{3}\pi r^3$, $4\pi r^2$) plus the cylinder-body terms ($\pi r^2 a$, $2\pi r a$), added together exactly once, with a = 0 as the clean boundary case where the body disappears and only the sphere remains.