Sectors and Arcs

A slice of the pizza

Cut a pizza across the middle and hand out slices. Each slice is a sector — a "pizza slice" of the circle. The two straight sides are radii, and the curved crust along the edge is the arc. The whole idea of this page hangs on one delightfully simple fact:

a sector is just a fraction of the whole circle.

A full turn around the centre is 360^\circ. If your slice opens out an angle \theta ("theta") at the centre, then it is exactly the fraction \frac{\theta}{360} of the pizza. Half the pizza is \frac{180}{360} = \tfrac12; a quarter is \frac{90}{360} = \tfrac14. Nothing more mysterious than that.

Two formulas, one idea

Because a sector is the fraction \frac{\theta}{360} of the circle, everything about it is that same fraction of the whole circle's version. Take that fraction of the circumference and you get the arc length (the length of the crust):

\text{arc length} = \frac{\theta}{360} \times 2\pi r

Take that same fraction of the area and you get the sector area (how much pizza is on the slice):

\text{sector area} = \frac{\theta}{360} \times \pi r^2

You do not have to memorise these as two separate spells. They are the one idea — "a fraction of the circle" — wearing two hats.

a sector is \frac{\theta}{360} of the whole circle;
arc length = \frac{\theta}{360} \times the circumference (2\pi r);
sector area = \frac{\theta}{360} \times \pi r^2;
a semicircle is \theta = 180^\circ (half); a quarter circle is \theta = 90^\circ.

See a sector on a circle

Step through the figure: two radii open out an angle \theta, then the slice between them is shaded — here a quarter of the circle.

Worked example 1 — arc length and area of a slice

A sector has angle \theta = 72^\circ in a circle of radius r = 5\text{ cm}. First get the fraction: \frac{72}{360} = \tfrac15 — this slice is one-fifth of the circle.

\text{arc length} = \tfrac15 \times 2\pi \times 5 = 2\pi \approx 6.28 \text{ cm}

\text{sector area} = \tfrac15 \times \pi \times 5^2 = 5\pi \approx 15.71 \text{ cm}^2

Notice the arc is a length (cm) and the area is a square measure (\text{cm}^2) — a handy check that you used the right formula.

Worked example 2 — the perimeter of a sector (the classic trap)

A sector has angle 90^\circ in a circle of radius r = 8\text{ cm}. Find its perimeter — the whole distance around the outside of the slice.

The edge of a slice is not only the curved crust. Walk right round it: along one straight radius, around the arc, then back along the other radius. So the perimeter is the arc plus two radii:

\text{arc} = \tfrac14 \times 2\pi \times 8 = 4\pi \approx 12.57 \text{ cm}

\text{perimeter} = \text{arc} + 2r = 4\pi + 2 \times 8 \approx 12.57 + 16 = 28.57 \text{ cm}

Answer the "perimeter" with just the arc and you drop a full 16\text{ cm} — which is exactly the mistake the next box is about.

Worked example 3 — working backward to find the angle

Sometimes you know the arc and want the angle. A circle has radius r = 6\text{ cm}, and one of its arcs is 3\pi \approx 9.42\text{ cm} long. What angle \theta does that arc subtend at the centre?

Start from the arc-length formula and rearrange for \theta:

\text{arc} = \frac{\theta}{360} \times 2\pi r \;\Rightarrow\; \theta = \frac{\text{arc}}{2\pi r} \times 360

\theta = \frac{3\pi}{2\pi \times 6} \times 360 = \frac{3\pi}{12\pi} \times 360 = \tfrac14 \times 360 = 90^\circ

The \pis cancel and out drops a clean 90^\circ — a quarter circle, just as the neat arc suggested.

The perimeter of a sector is arc + 2r, not just the arc. A sector has three edges: the curved arc and the two straight radii. Forgetting the two radii is the single most common sector-perimeter slip — always add 2r to the arc.
Use \frac{\theta}{360}, in degrees. These formulas take the angle as a fraction of a full turn, so \theta must be in degrees here. Don't reach for \frac{\theta}{180} or flip it to \frac{360}{\theta} — it is the slice's share of 360^\circ every time.

Everywhere you look, once you start noticing. A slice of pizza or pie is a sector. A windscreen wiper sweeps out a sector each time it swings. A Pac-Man mouth is a circle with a sector bitten away. And every wedge of a pie chart is a sector whose angle is that category's share of 360^\circ — one of the most instantly recognisable graphs in the world is built entirely from "fraction of the circle".

That fraction idea also opens a door. Instead of measuring angle out of 360^\circ, mathematicians often measure it in radians — where the angle is the arc length for a unit circle. In radians these very formulas get even simpler: the arc length becomes just r\theta, and the sector area becomes \tfrac12 r^2 \theta. Same slices, tidier algebra.