Mensuration
Areas related to circles
Find the arc length, area and perimeter of a sector, and the area of a segment.
Cut a circular cake and you get a sector: two radii and the arc between them. A sector is a fraction of the whole circle, and the angle at the centre says which fraction. A 90° sector is 90 out of 360, so a quarter. A 60° sector is a sixth. Once you have that fraction, every sector question is one multiplication away from a question about the whole circle.
The fraction is the whole idea
For an angle of θ at the centre, the fraction is θ/360. Area of sector = (θ/360) × πr². Arc length = (θ/360) × 2πr. Perimeter of the sector = arc + 2r, because two of its three edges are radii — the edge people forget to add twice.
Worked example
A sector of a circle of radius 21 has an angle of 60°. Take π = 22/7. Find its area.
Whole circle: (22/7) × 21 × 21 = 1386.
With π as 22/7, a radius that is a multiple of 7 makes the sevens cancel and the arithmetic stay whole.
A segment is different. Join the two ends of the arc with a straight chord, and the region between the chord and the arc is a segment — the sector with its triangle sliced off. So the area of a segment is the area of the sector minus the area of the triangle made by the two radii and the chord. Nothing new to memorise; just a subtraction in the right order.
Try it together
A sector of a circle of radius 10 has an angle of 90° at the centre. Take π = 3.14. The triangle made by the two radii and the chord has an area of 50. Let us find the minor segment.
Sector first, triangle second, subtract third.
1.What is 3.14 × 10 × 10?
Have a go
Have a go on your own: a sector of a circle of radius 7 cm has an angle of 180° at the centre. Take π = 22/7. What is its area, in square centimetres?
Hint: 180° is half a turn.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.