Arcs and Sectors

The angle of a sector

A sector is the slice of a circle between two radii and the arc that joins their ends. The angle between the two radii, at the center, is the sector's angle. A full turn about the center is 360°, so the angle says how much of the full turn the sector takes up.

A sector whose two radii are 90° apart: a quarter turn about the center.

A quarter of the circle

This sector turns 90° out of 360°, and $\frac{90}{360} = \frac{1}{4}$. Four sectors like it, side by side, make the whole circle. So the sector is a quarter of the circle: its arc is a quarter of the circumference, and its area is a quarter of the area of the circle.

For a circle with a radius of 12 cm, the circumference is $2\pi \times 12 = 24\pi$ cm, so the arc of the quarter is $24\pi / 4 = 6\pi$ cm. The area of the circle is $\pi \times 12^{2} = 144\pi$ cm², so the area of the quarter is $144\pi / 4 = 36\pi$ cm².

The quarter sector with its arc: a quarter of the area, and a quarter of the circumference.

Any angle: divide by 360

Cut a circle into 360 sectors, each with an angle of 1°. They are all the same shape and size, since each one is the next one turned by 1°, so each is $\frac{1}{360}$ of the circle. Write a sector's angle as $\theta$, the Greek letter theta. A sector with an angle of $\theta$ degrees is made of $\theta$ of these 1° sectors, so it is $\frac{\theta}{360}$ of the circle.

A 120° sector is $\frac{120}{360} = \frac{1}{3}$ of the circle. A 60° sector is $\frac{60}{360} = \frac{1}{6}$. A 45° sector is $\frac{45}{360} = \frac{1}{8}$. The angle does not have to give a simple fraction: a 50° sector is $\frac{50}{360} = \frac{5}{36}$ of the circle.

Three sectors of 120° make the whole circle, so each is a third of it.

Arc length and sector area

So the arc is $\frac{\theta}{360}$ of the circumference, and the sector is $\frac{\theta}{360}$ of the area of the circle. The arc length is $\frac{\theta}{360} \times 2\pi r$, and the sector area is $\frac{\theta}{360} \times \pi r^{2}$.

Take a sector with an angle of 80° in a circle of radius 9 cm. Its fraction of the circle is $\frac{80}{360} = \frac{2}{9}$. The circumference is $2\pi \times 9 = 18\pi$ cm, so the arc is $\frac{2}{9} \times 18\pi = 4\pi$ cm, which is 12.57 cm to two decimal places. The area of the circle is $\pi \times 9^{2} = 81\pi$ cm², so the sector's area is $\frac{2}{9} \times 81\pi = 18\pi$ cm², which is 56.55 cm² to two decimal places.

Leaving the answers as $4\pi$ and $18\pi$ keeps them exact. Change them to decimals only at the end, when a decimal is asked for.

The perimeter of a sector

The perimeter of a sector is the whole way round its edge: the arc and the two radii. For the 80° sector of radius 9 cm, the perimeter is $4\pi + 9 + 9 = 4\pi + 18$ cm, which is 30.57 cm to two decimal places.

Finding the angle

The fraction also works backwards. A sector of a circle of radius 10 cm has an area of $35\pi$ cm². The whole circle has an area of $\pi \times 10^{2} = 100\pi$ cm², so the sector is $35\pi / 100\pi = \frac{35}{100} = \frac{7}{20}$ of the circle. Its angle is $\frac{7}{20} \times 360^\circ = 126^\circ$.

The usual mistakes

Mixing up the arc and the area. The arc is a length, a fraction of $2\pi r$, measured in units such as cm. The area is a fraction of $\pi r^{2}$, measured in square units such as cm².

Stopping at the fraction. $\frac{2}{9}$ is how much of the circle the sector is; the arc length is $\frac{2}{9}$ of the circumference.

Leaving out the radii from the perimeter. The arc alone is only the curved edge; a sector also has two straight edges.

Dividing by 180 instead of 360. A full turn is 360°, so a 90° sector is $\frac{90}{360} = \frac{1}{4}$ of the circle, not $\frac{1}{2}$.