Stadium Shape Calculator
Calculate area and perimeter of a geometric stadium shape from any 2 known values
Results
| Property | Symbol | Value |
|---|
Stadium Shape Formulas
| Property | Formula | Notes |
|---|---|---|
| Area (A) | A = πr² + 2ra | Circle area + rectangle area |
| Perimeter (P) | P = 2(πr + a) | Full circle circumference + 2 sides |
| Side from r, A | a = (A − πr²) / (2r) | Rearranged area formula |
| Side from r, P | a = P/2 − πr | Rearranged perimeter formula |
| Radius from a, P | r = (P/2π) − (a/π) | Rearranged for r |
What Is a Stadium Shape?
A stadium shape in geometry is a two-dimensional figure formed by taking a rectangle and attaching a semicircle to each of the two shorter ends. The result looks like an athletics running track viewed from above, or like a capsule or pill. This shape is also called a discorectangle, an oblong, or a Bunimovich stadium in mathematics.
The stadium shape is defined by two measurements: the radius r of the semicircular ends, and the side length a of the rectangular middle section. When a = 0, the shape reduces to a perfect circle. When r is very small relative to a, the shape approaches a rectangle.
Real World Applications
Stadium shapes appear frequently in architecture, sports facility design, and manufacturing. Running tracks, drug capsules, racing circuits, and the cross-sections of certain pipes all follow the stadium geometry. In physics, the Bunimovich stadium is famous for demonstrating chaotic billiard ball dynamics.
Area Formula Explained
The area formula adds the area of a full circle (πr²) to the area of the central rectangle (2r × a = 2ra). The rectangle has width 2r (equal to the diameter) and length a. This additive approach makes the formula easy to derive and remember.
Frequently Asked Questions
A stadium shape is a 2D geometric figure made up of a rectangle with a semicircle attached to each short end. It looks like a running track, a capsule, or a pill. It is formally known as a discorectangle. The two key measurements are the radius r of each semicircle and the length a of the rectangular side connecting them.
The area of a stadium shape is A = πr² + 2ra, where r is the radius of the two semicircular ends and a is the length of the straight rectangular section. This adds the area of one full circle (πr²) to the area of the rectangle in the middle (2ra, since the rectangle has width 2r and length a).
The perimeter of a stadium is P = 2(πr + a). The two straight sides together contribute 2a to the perimeter, and the two semicircular ends together form one full circle with circumference 2πr. Adding these gives P = 2a + 2πr = 2(a + πr).
When the side length a = 0, the stadium shape becomes a perfect circle with radius r. The area formula A = πr² + 2r(0) = πr² and the perimeter formula P = 2(πr + 0) = 2πr both reduce to the standard circle formulas. The stadium is therefore a generalization of the circle.
A stadium shape is also known as a discorectangle, an oblong, a pill shape, or a Bunimovich stadium. The name discorectangle comes from combining disc (referring to the circular ends) with rectangle (the central section). In physics and mathematics, it is often called a Bunimovich stadium after the mathematician Leonid Bunimovich.