Question

The following figure shows a regular octagon inscribed in a circle. The arc length from $A$ to $B$ is $6\pi$. What is the area of the shaded region of the circle?

• A. $8\pi$
• B. $16\pi$
• C. $24\pi$
• D. $36\pi$
• E. $64\pi$

Answer 1

The correct option is C $24\pi$

As the octagon is inscribed in a circle, we understand that as the total angle at the centre of the circle is $2\pi$ and is divided into $8$ equal parts by the octagon, the angle from $A$ to $B$ is three parts.

This means angle of the sector from $A$ to $B$ is $\frac{3}{8}\times 2\pi =\frac{3\pi }{4}={135}^0$
Length of an arc subtending an angle $\theta =\frac{\theta }{360}\times 2\pi R$ where $R$ is the radius of the circle.
$\Rightarrow 6\pi =\frac{135}{360}\times 2\pi R$
$\Rightarrow R=8$
Area of a sector of a circle of radius 'R' and angle $\theta =\frac{\theta }{360}\pi R^2$
Hence, area of the sector of the circle of radius $8$ cm and angle ${135}^0=\frac{135}{360}\times \pi \times 8\times 8=24\pi$