The Length of an Arc

Recall, the circumference of a circle is 2πr. In the diagram below, the arc length l or XY subtends an angle θ° at the centre O with radius r.

In general, the length of an arc of a circle is proportional to the angle at which the arc subtends at the centre.

Therefore, the arc length “l” or |XY| is given as:

\(\scriptsize l = \normalsize \frac {\theta}{360}\scriptsize \: \times \: 2 \pi r \)

Example 8.1.1:

In terms of π, what is the length of an arc which subtends an angle of 30º at the centre of a circle of radius \( \scriptsize 3\normalsize \frac {1}{2} \: \scriptsize cm \)?

Solution:

Using |XY| = \( \frac {\theta}{360}\scriptsize \: \times \: 2 \pi r \)

θ = 30º

radius = \(\scriptsize 3\frac {1}{2} = \normalsize \frac {7}{2}\)

|XY| = \( \frac {30}{360}\scriptsize \: \times \: 2 \pi \: \times \: \normalsize \frac{7}{2} \)

|XY| = \( \scriptsize 7 \normalsize \frac{\pi}{12} \)

|XY| = \(\frac{7 \pi}{12} \)

Example 8.1.2:

What angle does an arc 5.5 cm in length subtend at the centre of a circle of diameter 7 cm? (WAEC)

Solution:

Using |AB| = \( \frac {\theta}{360}\scriptsize \: \times \: 2 \pi r \)

• AB = 5.5 cm
• r = 7 cm
• θ = ?
• \( \scriptsize \pi = \normalsize \frac{22}{7} \)

5.5 = \( \frac {\theta}{360}\scriptsize \: \times \: 2 \: \times \: \normalsize \frac{22}{7} \scriptsize \: \times \: 7 \)

⇒ \( \frac {11}{2} = \frac {44\theta}{360} \)

⇒ \( \frac {11}{2} = \frac {11\theta^{\circ}}{90}\)

⇒ θ = \( \frac {11 \: \times \: 90}{2 \: \times \: 11}\)

⇒ θ = \( \frac {990}{22}\)

⇒ θ = 45°

Example 8.1.3:

An arc length of 28 cm subtends an angle of 24º at the centre of a circle. In the same circle,