Question

Given that a chord with length $l$ subtends an angle $\theta$ at the centre of the circle with diameter $d$ and radius $r$, find the perimeter of the segment.

• A. $\frac{\theta }{{360}^{\circ }}\times \pi r+l$
• B. $\frac{\theta }{{360}^{\circ }}\times \pi d+l$
• C. $\frac{\theta }{{180}^{\circ }}\times \pi r+l$

Answer 1

The correct option is C $\frac{\theta }{{180}^{\circ }}\times \pi r+l$

Since the chord subtends an angle $\theta$ at the centre of the circle, its perimeter must be the sum of the length of the chord $l$ and the length of the arc.
$\text{Perimeter}=\frac{\theta }{{360}^{\circ }}\times \pi d+l$
$\Rightarrow \text{Perimeter}=\frac{\theta }{{360}^{\circ }}\times \pi \left(2r\right)+l$
$\therefore \text{Perimeter}=\frac{\theta }{{180}^{\circ }}\times \pi r+l$