Stadium Calculator: Area and Perimeter of a Pill Shape

What a stadium shape is

Take a circle, slice it in half straight through the center, and pull the two halves apart. Insert a rectangle of length $a$ between them, matching the circle's diameter as the rectangle's height. The result — a running track, a pill capsule, a swimming lane — is called a stadium (or pill, capsule, discorama).

$A = \pi r^2 + 2ra \qquad P = 2(\pi r + a)$

Why these formulas work

The area comes from two pieces: the two semicircular caps rejoined make one full circle, contributing $\pi r^2$; the rectangular middle section contributes $2r \times a$ (height $2r$, width $a$). Adding them gives $A = \pi r^2 + 2ra$.

The perimeter also comes from two pieces: the two semicircular arcs rejoined make one full circumference, $2\pi r$; the two straight sides each contribute length $a$, for $2a$ total. Adding them gives $P = 2\pi r + 2a = 2(\pi r + a)$.

Worked example: radius r = 5, length a = 10

$A = \pi(5^2) + 2(5)(10) = 25\pi + 100 \approx 178.5398$ $P = 2(\pi \cdot 5 + 10) = 2(5\pi + 10) \approx 51.4159$

Because both the radius and length are exact here, the area also has an EXACT symbolic form ($25\pi + 100$) alongside the decimal approximation.

Solving in the other direction

Given the radius plus either the area or the perimeter, this calculator solves for the rectangle length first, then derives the rest.

Worked example: radius r = 5, area A = 178.5398

Rearranging the area formula gives $a = (178.5398 - \pi(25))/10 = (178.5398 - 78.5398)/10 = 10$. The perimeter follows as $P = 2(\pi(5) + 10) \approx 51.4159$.

Common mistakes

• Forgetting the rectangle entirely. Computing only $\pi r^2$ (the circle part) and skipping $2ra$ under-counts the area — a stadium is always bigger than the circle its caps came from (unless $a = 0$).
• Using $r$ instead of $2r$ for the rectangle's height. The rectangle spans the caps' full diameter, not just the radius.
• Confusing perimeter with area. Perimeter is a length ($2\pi r + 2a$); area is a surface ($\pi r^2 + 2ra$) — they don't share units.

Where this shows up

• Running tracks and athletics fields: the classic 400m oval track is a stadium shape, which is where the name comes from.
• Pills and capsules: medicine capsules are 3D versions (a capsule, the solid of revolution of a stadium) built from the same two-piece geometry.
• Swimming lanes and rounded tables: any rectangle with semicircular rounded ends — a rounded rectangle taken to its full curve — is a stadium.

A stadium is the simplest shape built from two others: know the circle's radius and the rectangle's length, and every other measure — area, perimeter, or either one worked backward — follows directly.