Areas Related to Circles

Why This Matters

A clock’s minute hand sweeps a slice of a circle every few minutes. A car’s wiper cleans a curved patch of glass. A horse tied to a peg can graze only a pie-shaped piece of a field. A lighthouse warns ships across a fan of sea. Every one of these is the same shape — a slice of a circle — and every one of these is a real area someone needs to measure.

You already know how to find the area of a whole circle (πr²) and its boundary (2πr). But the world rarely hands you whole circles. It hands you slices: a wedge between two radii (a sector), or the curved cap between a straight chord and the arc above it (a segment).

This chapter is delightfully short, because it rests on one honest idea: a slice is just a fraction of the whole circle. Cut a 360° pizza into a 60° wedge and you’ve taken 60/360 = one-sixth of it — one-sixth of the area, one-sixth of the crust. Get that fraction idea, and every formula here writes itself.

The Big Idea

A sector is the wedge between two radii and the arc joining them; its angle at the centre is θ. A circle is really a sector of angle 360°, so a sector of angle θ is just the fraction θ/360 of the whole circle. That single fraction gives both its area, (θ/360) × πr², and its arc length, (θ/360) × 2πr. A segment is the region between a chord and its arc; you get it by taking the sector and cutting away the triangle the two radii make — segment = sector − triangle.

Let’s Break It Down

Recall: the whole circle (area and circumference)

Before slicing, fix the two facts about the whole circle of radius r:

• Circumference (the distance round the boundary) = 2πr
• Area (the flat region inside) = πr²

Here π (pi) is the constant ≈ 3.14159…; in this chapter we use π = 22/7 unless a problem says otherwise (sometimes π = 3.14). Keep them apart: circumference is a length (cm, m) so it has powers of 1; area is a region (cm², m²) so it carries the square. If a problem gives the diameter d, remember r = d/2 — halve it before substituting.

Sector: the area and the arc length (and why θ/360)

A sector is the part of the disc enclosed by two radii and the arc between their ends — picture OAPB, a wedge with its sharp point at the centre O and its curved edge on the circle. The angle ∠AOB at the centre is the angle of the sector, written θ.

Tap to zoom

Why the fraction θ/360 works — this is the whole chapter in one move. Go all the way around the centre and you turn through 360°, and that full turn sweeps out the entire circle: area πr², boundary 2πr. So one full circle “costs” 360° of angle. Now ask: what does 1° sweep? Exactly 1/360 of the circle. And θ degrees sweep θ times that — θ/360 of the circle. That is the unitary method: find the value of one unit (1°), then scale up to θ. So:

• Area of a sector of angle θ = (θ/360) × πr²
• Length of its arc = (θ/360) × 2πr

Same fraction, two different “wholes”: for area the whole is πr², for arc length the whole is the circumference 2πr. Notice that when θ = 360 the fraction is 1 and you get back πr² and 2πr — the formulas contain the whole circle as a special case, which is a good sanity check.

The leftover bigger wedge is the major sector, with angle 360° − θ. You can find its area the same way (fraction (360 − θ)/360), or simply as major sector = πr² − minor sector — the two pieces fill the whole circle.

Segment: sector minus triangle

A segment is the region trapped between a chord and the arc above it — the curved “cap” you’d cut off with one straight knife-stroke across the circle. The chord AB and the arc APB enclose it.

How do you find its area? Look carefully: if you take the sector OAPB (the wedge) and remove the triangle OAB (the straight-sided part with O as a vertex), what’s left is exactly the cap between chord and arc — the segment.

Tap to zoom

So the formula is simply:

Area of segment = Area of sector − Area of triangle OAB = (θ/360) × πr² − area of △OAB

The only new work is the triangle’s area. The triangle has two sides equal to r (the radii) with the angle θ between them; its area is (1/2) × r² × sin θ, or you can drop a perpendicular from O to the chord and use base × height ÷ 2 (the perpendicular bisects the chord). For the friendly exam angles (60°, 90°, 120°) the triangle works out neatly.

Just like sectors, a chord creates a minor segment (the small cap) and a major segment (the rest); and major segment = πr² − minor segment.

Common Mistakes

Quick Check

Practice Problems

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Medium

Challenge

Summary

You should now be able to explain:

• For a whole circle of radius r: circumference = 2πr (a length) and area = πr² (a region). Use π = 22/7 or 3.14; if given the diameter, halve it to get r.
• A sector is the wedge between two radii and the arc; its central angle is θ.
• A sector is the fraction θ/360 of the whole circle, because a full turn (360°) sweeps the whole circle — so 1° sweeps 1/360 of it and θ° sweeps θ/360.
• Area of a sector = (θ/360) × πr²; length of its arc = (θ/360) × 2πr — same fraction, but multiplied by the area or the circumference respectively.
• The major sector/segment is just the whole circle minus the minor one.
• A segment is the region between a chord and its arc, and segment area = sector area − triangle OAB area (subtract area from area, never lengths).

What’s Next

So far everything has stayed flat — areas of regions you can draw on paper. Next, in Surface Areas and Volumes, those shapes lift off the page into three dimensions: cubes, cylinders, cones and spheres. You’ll find the surface area (the skin you’d paint) and the volume (the space inside), and you’ll combine solids — a cone on a cylinder, a hemisphere on a cube — exactly the way the circle slices here combined into bigger shapes.