Mathematics
Sector of a Circle: The Slice Between Two Radii
A sector is bounded by two radii and an arc. Its area and arc length are both the angle over 360 of the whole circle — worked on a 72° slice.
Sector
A sector is the region of a circle enclosed by two radii and the arc between their endpoints, like a slice cut from a whole pizza.
- Where students meet it
- Grade 8 mathematics, and again at GCSE and IGCSE where sector area and arc length appear in compound-shape and perimeter questions.
The short answer
A sector of a circle is the region enclosed by two radii and the arc between them, the shape of a slice cut from a whole pizza. Its area and its arc length are both found by taking the fraction of the circle the centre angle covers: a 72-degree sector is exactly one fifth of the circle.
An example
Area = (72 ⁄ 360) × π × 10² = 20π ≈ 62.8 cm²
A sector of a circle of radius 10 cm has a centre angle of 72 degrees. Since 72 out of 360 is one fifth, the sector is one fifth of the circle's 100π cm² area, which is 20π or about 62.8 cm². The same fifth applied to the circumference of 20π cm gives an arc of 4π cm, roughly 12.6 cm.
Two radii and an arc
A sector has its point at the centre. That is what separates it from a segment, which is cut off by a chord and never touches the centre at all. If the two straight edges are radii, you have a sector.
Any pair of radii creates two sectors at once. The smaller is the minor sector and the larger the major sector, and their angles add to 360 degrees. A question asking for the major sector of a 72-degree slice is asking about the remaining 288 degrees, so the fraction to use is 288 over 360.
Everything is the angle over 360
There is really only one idea on this topic. Work out what fraction of a full turn the centre angle represents, then take that fraction of whatever you want — the area of the circle for a sector area, the circumference for an arc length.
Written as formulas that is area = (θ/360) × πr² and arc = (θ/360) × 2πr, but remembering them as fractions makes the arithmetic easier and the errors more visible. A 90-degree sector should come out as a quarter, and if it does not, something has gone wrong before the formula was even reached.
A handful of angles come up often enough that the fraction is worth recognising on sight rather than working out each time:
- 30° is one twelfth of the circle
- 45° is one eighth
- 60° is one sixth
- 72° is one fifth
- 90° is one quarter
- 120° is one third
- 180° is a semicircle
The perimeter of a sector is not its arc
The boundary of a sector has three parts: the arc and both radii. For the 72-degree sector of a radius-10 circle, that is 4π + 10 + 10, about 32.6 cm. Answering 12.6 cm gives only the curved part and leaves the shape open.
The same care applies to the wording. Arc length means the curve alone, perimeter means the whole way round, and questions use both. Reading which one is wanted takes a second and is worth a mark.
Common questions
What is the difference between a sector and a segment?
A sector is bounded by two radii and an arc, so its point sits at the centre of the circle. A segment is bounded by a chord and an arc and does not reach the centre. Draw both on the same circle once and the difference stops being confusing.
How do you find the angle of a sector when the area is given?
Divide the sector area by the whole circle's area to get the fraction, then multiply by 360. A sector of 20π cm² in a circle of radius 10 cm covers 20π out of 100π, which is one fifth, and one fifth of 360 degrees is 72 degrees.
What is a semicircle in these terms?
A sector with a centre angle of 180 degrees, where the two radii lie in a straight line and form a diameter. It is also a segment, since that diameter is a chord. It is the one shape that is genuinely both, which is why it is a poor example to learn the difference from.
Should the answer be left in terms of π?
Leave it as a multiple of π unless the question asks for a decimal or specifies a number of significant figures. Writing 20π cm² is exact, and it stays exact if the value is needed in later working. A non-calculator paper almost always expects the π form.
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Knowing the word is not the same as using it
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