Arcs, and a 400 m track

A circle of diameter d has circumference πd, or 2πr.

Pieces of a circle:

• a semicircle is half of it: πr (reflect in a diameter and the two halves swap, so they are equal);
• a quarter circle is a quarter: πr/2 (turn by 90° and each quarter moves onto the next).

Any arc, by the unitary method. 360° gives the whole circumference 2πr, so 1° gives 2πr/360, and θ° gives 2πr × θ/360. A semicircle is θ = 180; a quarter circle, θ = 90.

A real 400 m track has two straights of 84.39 m and two semicircular bends; the inner bend has radius 36.5 m. A runner 0.3 m from the inner edge runs:

• straights: 2 × 84.39 = 168.78 m;
• both bends, a full circle of radius 36.8 m: 2 × 3.1416 × 36.8 = 231.22 m;
• total: 400 m.

The stagger, worked out. Each lane is 1.22 m wide, so the next lane's bends have a radius 1.22 m bigger. Over the two bends (a full circle) that adds 2π × 1.22 ≈ 7.67 m. Every pair of neighbouring lanes differs by the same amount.