Question

Find the area of a segment of a circle with a central angle of $135$ degrees and a radius of $2$. Express answer to nearest integer.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$A_{segment}=A_{sector}-A_{triangle}$
$A_{sector}=\frac{n}{360}\pi r^2$
$=\frac{135}{360}\pi 2^2$
= 4.71 $c{m}^2$
$A_{triangle}=\frac{1}{2}ab\left(\sin \theta \right)$

Where $\theta$ is the angle between the sides of lengths a and b
b = 2
h = 2

$\theta ={135}^o$
$A_{triangle}=\frac{1}{2}ab\sin \left({135}^o\right)$
= $\frac{1}{2}2\times 2\times \frac{1}{\sqrt{2}}$
= $1.41$
$A_{segment}=4.71-1.41=3.3=3$