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}$90/360 = 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$2π × 12 = 24π cm, so the arc of the quarter is $24\pi /4=6\pi$24π / 4 = 6π cm. The area of the circle is $\pi \times 12^2=144\pi$π × 12² = 144π cm², so the area of the quarter is $144\pi /4=36\pi$144π / 4 = 36π 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}$1/360 of the circle. Write a sector's angle as $\theta$θ, the Greek letter theta. A sector with an angle of $\theta$θ degrees is made of $\theta$θ of these 1° sectors, so it is $\frac{\theta }{360}$θ/360 of the circle.

A 120° sector is $\frac{120}{360}=\frac{1}{3}$120/360 = 1/3 of the circle. A 60° sector is $\frac{60}{360}=\frac{1}{6}$60/360 = 1/6. A 45° sector is $\frac{45}{360}=\frac{1}{8}$45/360 = 1/8. The angle does not have to give a simple fraction: a 50° sector is $\frac{50}{360}=\frac{5}{36}$50/360 = 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 $\frac{\theta }{360}$θ/360 of the circumference, and the sector is $\frac{\theta }{360}$θ/360 of the area of the circle. The arc length is $\frac{\theta }{360}\times 2\pi r$θ/360 × 2πr, and the sector area is $\frac{\theta }{360}\times \pi r^2$θ/360 × πr².

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}$80/360 = 2/9. The circumference is $2\pi \times 9=18\pi$2π × 9 = 18π cm, so the arc is $\frac{2}{9}\times 18\pi =4\pi$2/9 × 18π = 4π cm, which is 12.57 cm to two decimal places. The area of the circle is $\pi \times 9^2=81\pi$π × 9² = 81π cm², so the sector's area is $\frac{2}{9}\times 81\pi =18\pi$2/9 × 81π = 18π cm², which is 56.55 cm² to two decimal places.

Leaving the answers as $4\pi$4π and $18\pi$18π 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$4π + 9 + 9 = 4π + 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$35π cm². The whole circle has an area of $\pi \times 10^2=100\pi$π × 10² = 100π cm², so the sector is $35\pi /100\pi =\frac{35}{100}=\frac{7}{20}$35π / 100π = 35/100 = 7/20 of the circle. Its angle is $\frac{7}{20}\times 360^{\circ }=126^{\circ }$7/20 × 360° = 126°.

The usual mistakes

Mixing up the arc and the area. The arc is a length, a fraction of $2\pi r$2πr, measured in units such as cm. The area is a fraction of $\pi r^2$πr², measured in square units such as cm².

Stopping at the fraction. $\frac{2}{9}$2/9 is how much of the circle the sector is; the arc length is $\frac{2}{9}$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}$90/360 = 1/4 of the circle, not $\frac{1}{2}$1/2.