Question

In the given figure, O is the centre of circle. XYZ is an arc of the circle subtending an angle of $45^\circ$ at the centre. If the radius of the circle is $32 \text{ cm}$, then the length of the arc XYZ is :

• A. $4\pi \text{ cm}$
• B. $8\pi \text{ cm}$
• C. $64\pi \text{ cm}$
• D. $128\pi \text{ cm}$

Hint:

An angle of $45^\circ$ represents $\frac{45}{360} = \frac{1}{8}\text{th}$ of the entire circumference.
The total circumference of the circle is $2\pi r = 2\pi(32) = 64\pi \text{ cm}$.
The arc length is simply $\frac{64\pi}{8} = 8\pi \text{ cm}$.

Answer 1

The Correct Option is B

Step 1: Find the full circumference first.
For radius $32$ cm, the whole circumference is $2\pi r = 2\pi(32)=64\pi$ cm.
Step 2: Work out what fraction of the full circle the arc represents.
An angle of $45^\circ$ out of the full $360^\circ$ is $\frac{45}{360}=\frac18$ of one complete revolution.
Step 3: Take that fraction of the circumference.
Arc length $=\frac18\times64\pi = 8\pi$ cm.
\[ \boxed{8\pi \text{ cm}} \]