Question

In the figure above, $AB$ and $DC$ are arcs of concentric circles with centre $O$. The perimeter of figure $ABCD$ is $22$ cm. Calculate the area of the fig. $ABCD$.

• A. $20c{m}^2$
• B. $10c{m}^2$
• C. $40c{m}^2$
• D. $30c{m}^2$

Answer 1

The correct option is D $30c{m}^2$

Length of arc of a circle is given by $r\theta$

Let r1 = 2cm be the radius of inner circle

Let r2 = 8cm be the radius of outer circle

$AB=r_1\theta =2\theta$

$DC=r_2\theta =8\theta$

$AD=BC=6cm$

$P=AB+CD+AD+BC=22cm$

$⟹10\theta +12=22cm$

$\theta =1$

Area of sector is given by $1/2r^2\theta$

Area of $ABCD=$ Area of sector $CD$ – Area of sector $AB$

$=\frac{1}{2}{r}_2^2\times 1=\frac{1}{2}{r}_1^2\times 1$

$=\frac{1}{2}(8^2–2^2)=30c{m}^2$