Sectors and arcs

Two shapes cut from a circle:

The smaller piece is the minor sector (or segment), and the rest of the circle is the major one. If the minor sector has angle θ at the centre, the major sector has 360° – θ. If a question just says 'sector' or 'segment', it means the minor one.

Where the formulas come from (the unitary method):

• the whole circle is 360°, with area πr²
• so 1° gives an area of πr²/360
• so θ° gives an area of (θ/360) × πr²

The arc length works the same way, starting from the whole circumference 2πr:

So 'length → use 2πr; area → use πr²', and take the θ/360 fraction of it.

Major sector = the whole circle – the minor sector = πr² – (θ/360)πr².

Clocks. The minute hand goes round 360° in 60 minutes, so it turns 6° every minute. In 5 minutes it sweeps 30°, which is 1/12 of the circle. A car's wiper and a swinging pendulum also sweep sectors, and the length of the hand, wiper or pendulum is the radius.

A horse tied at a corner of a square field can reach a quarter circle, because the corner angle is 90°. Picture yourself tied to that corner to see what you can reach.

Perimeter of a sector = arc + two radii: don't forget the straight edges.