Find the area of the shaded region in the diagram, in terms of $\theta$

Question

Problem
A company is designing a new logo. The logo is created by removing two equal segments from a rectangle, as shown in the following diagram

The rectangle measures $5cm$ by $4cm$. The points $A$ and $B$ lie on the circle, with center $O$ and radius $2cm$ such that $\angle AOB=\theta$ where $0<\theta<\pi$, as shown here

• Find the area of one of the shaded segments [in pink] in terms of $\theta$.
• Given that area of the logo is $13.4 cm^2$, find the value of $\theta$.

My Approach
I have calculated the area in terms of $\theta$ and $\sin\theta$ but don't know how to solve for $\theta$ afterwards. There must be way to find the area of the removed segment without $\sin\theta$ and $\theta$. Can someone please guide for the same?

Answer 1

This can only be solved using a graphing tool which is allowed on the exam that this question was asked. The answer to this is $\theta = 135.05^{\circ}$

Answer 2

Working in radians, sector $AOB=\frac{\pi r^2\theta}{2\pi}=2\theta$

And since $\sin (2\theta)=2\sin\theta\cos\theta$, and radius $OA=2$, then

$\triangle AOB=\frac{1}{2}AB \cdot OC=AC \cdot OC=4\sin\frac{\theta}{2}\cos\frac{\theta}{2}=2\sin\theta$

Hence the $3.3\mathrm{cm^2}$ segment, expressed in terms of $\theta$ is$$2\theta-2\sin\theta$$And consulting a trigonometric table or calculator (if that is allowed here) shows that $2\theta-2\sin\theta=3.298$ for $\theta=135^o=2.356194\mathrm{rad}$