Areas with circles and copies

Wheels again: a 60 cm wheel moves π × 60 = 1320/7 cm a turn, so 100 turns cover about 188.57 m.

Scaling. Double every length and the area becomes 4 times (2 × 2); triple, and it becomes 9 times. You can tile the bigger shape with 4 (or 9) copies of the small one. Twice the size is not twice the paint.

Circles in a row inside a rectangle always cover π/4 of it, however many there are: each circle sits in its own square.

Trapezium area = ½(a + b)h, shown three ways:

• two copies make a parallelogram;
• a diagonal splits it into two triangles;
• cut and turn the top half.

Kite area = ½ × d₁ × d₂, because its diagonals are perpendicular.

Shared parts cancel. If two shapes have equal areas and share a common part, the leftovers are equal. That proves the regions in Figs. 6.49 and 6.55 equal, and that the lunes of Fig. 6.52 add up to the triangle.