Question

In Figure 5, rectangle $ABCD$ is inscribed in a circle. If the radius of the circle is $1$ and $AB=1$, find the area of the shaded region.

• A. $0.091$
• B. $0.285$
• C. $0.614$
• D. $0.705$
• E. $0.732$

Answer 1

Answer: (C)

$0.614$

Area of circle is $\pi (1{)}^2=\pi =3.143$
Given, $AB=1$ and angle made by $AB$ with center of circle is $60$ degrees, therefore angle made by $AD$ with center of circle is $120$ degrees.
Therefore $AD=\sqrt{3}\times 1=\sqrt{3}$
Area of rectangle is $1\times \sqrt{3}=1.732$
Area made by an arc $AD$ with center is $\frac{1}{3}\times 3.143=1.0476$
Area made by line $AD$ with center is $\frac{1}{2}\times 1\times 1\times \sin 120=0.433$
Therefore, area of shaded region is $1.0476-0.433=0.614$.