Perimeter puzzles

Break a shape into pieces: straight parts and arcs. Then add only the pieces on the outside border. Always ask: "Do the straight edges count, or only the curves?"

Example 1. Two circles of radius r, each passing through the other's centre. Joining the centres and a crossing point makes an equilateral triangle, so each inner arc is 120°, a third of its circle. The outline is two arcs of two thirds each: 2 × (2/3) × 2πr = 8πr/3.

Example 2. From P to Q, go along one big semicircle, or along three small semicircles on PQ. The big one is π × (its radius); the small ones are π × (sum of their radii). Both radius totals are half of PQ, so the two paths are equal.

A shortcut: semicircle lengths add like their diameters. In many figures you can add up the diameters and multiply by π/2.

Wheels. A wheel moves forward one circumference for each turn. A 56 cm tyre moves 176 cm a turn, so in 10 km (1 000 000 cm) it turns about 5682 times.