The length of an arc

A slice of the circle

Cut a round pizza straight from the middle to the edge, twice. The slice has a curved edge, which is the crust. That crust is an arc of the circle. The two cuts are radii, and together with the arc they bound a sector.

• Two points on a circle cut it into two arcs. The shorter one is the minor arc and the longer one is the major arc.
• The angle between the two radii is the angle of the arc at the centre. We call it $\theta$ (theta).
• A sector is the region inside the two radii and the arc.

How long is an arc?

A full turn at the centre is $360^\circ$, and it uses the whole circumference $2\pi r$. A slice with an angle of $90^\circ$ is $\dfrac{90}{360} = \dfrac{1}{4}$ of the turn, and its arc is $\dfrac{1}{4}$ of the circumference. We use the same fraction for every angle.

Example 1. A circle has a radius of $21\ \text{cm}$. Find the length of an arc that makes $60^\circ$ at the centre. Take $\pi = \dfrac{22}{7}$. Then give the exact value in terms of $\pi$.

1. The circumference. $2 \times \dfrac{22}{7} \times 21 = 132\ \text{cm}$.
2. The fraction. $\dfrac{60}{360} = \dfrac{1}{6}$.
3. The arc. $\dfrac{1}{6} \times 132 = 22\ \text{cm}$.
4. The exact value. $\dfrac{1}{6} \times 2\pi \times 21 = 7\pi\ \text{cm}$. A calculator gives $21.99\ \text{cm}$.

Example 2, the angle from the arc. An arc of length $11\ \text{cm}$ belongs to a circle of radius $14\ \text{cm}$. Find the angle at the centre. Take $\pi = \dfrac{22}{7}$.

1. The circumference. $2 \times \dfrac{22}{7} \times 14 = 88\ \text{cm}$.
2. The fraction. $\dfrac{11}{88} = \dfrac{1}{8}$ of the circumference.
3. The angle. We want $\theta$, so $\theta = \dfrac{1}{8} \times 360^\circ = 45^\circ$.

Example 3, an angle given as a ratio. An arc is $\dfrac{1}{9}$ of the whole circumference. The angle at the centre is $\dfrac{1}{9}$ of $360^\circ$, which is $40^\circ$. You do not need the radius.

Example 4, the major arc. In Example 1 the minor arc is $22\ \text{cm}$. The whole circumference is $132\ \text{cm}$, so the major arc is $132 - 22 = 110\ \text{cm}$. Its angle is $360^\circ - 60^\circ = 300^\circ$.

The perimeter of a sector

The boundary of a sector has three parts: the arc and the two radii. So

Example 5. A sector of a circle of radius $14\ \text{cm}$ has an angle of $90^\circ$. The arc is $\dfrac{1}{4} \times 88 = 22\ \text{cm}$. The perimeter is $22 + 14 + 14 = 50\ \text{cm}$.

Do not forget the two radii. A perimeter of $22\ \text{cm}$ would be wrong, because that is the arc alone.

Radians (A-Level and IB)

In the Indian syllabus the angle of an arc is in degrees. A-Level and IB use radians. An angle of $\theta$ radians gives $l = r\theta$. A full turn is $2\pi$ radians, so $360^\circ = 2\pi$ radians and $60^\circ = \dfrac{\pi}{3}$ radians. For Example 1, $l = 21 \times \dfrac{\pi}{3} = 7\pi\ \text{cm}$. It is the same answer, because $\dfrac{\theta}{360} \times 2\pi r$ and $r\theta$ are the same rule written two ways.

Practise with new numbers

Try it in Python

This program finds the arc and the perimeter of the sector for six angles in a circle of radius $21\ \text{cm}$. It uses $\pi = \dfrac{22}{7}$ because the question says so.

Common mistakes

• Using the area formula for a length. The arc length uses $2\pi r$. The area uses $\pi r^2$.
• Forgetting the fraction. An arc is only a part of the circumference. Multiply by $\dfrac{\theta}{360}$.
• Leaving out the radii in the perimeter of a sector.
• Using degrees in $l = r\theta$. That short rule needs the angle in radians.

Remember these

• $l = \dfrac{\theta}{360} \times 2\pi r$ for an angle in degrees, and $l = r\theta$ for an angle in radians.
• The perimeter of a sector is $l + 2r$.
• The major arc is the circumference minus the minor arc.
• Use $\pi = \dfrac{22}{7}$ only when told, and show the exact form with $\pi$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. An arc makes $45^\circ$ at the centre of a circle of radius $28\ \text{cm}$. Take $\pi = \dfrac{22}{7}$. Find its length.

2. An arc of length $11\ \text{cm}$ belongs to a circle of radius $7\ \text{cm}$. Take $\pi = \dfrac{22}{7}$. Find the angle at the centre.

3. An arc makes $150^\circ$ at the centre of a circle of radius $12\ \text{cm}$. Write its length in terms of $\pi$.

4. An arc is one third of the circumference of a circle. What angle does it make at the centre?

5. A sector has a radius of $10\ \text{cm}$ and an angle of $36^\circ$. Write its perimeter in terms of $\pi$.