Question

$\huge\red{\boxed{\sf QuEStiON }}$

In the Figure, PQ and AB are respectively the arcs of two concentric circles of a radii 7 cm and 3.5 cm and centre O. If ∠POQ = 30°, then find the area of the shaded region

$\huge\blue{\boxed{\sf Note: }}$

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Answer 1

$\displaystyle\large\underline{\sf\red{Given}}$

✭ There are two concentric circles of radius 7 cm & 3.5 cm

✭ There is a sector drawn with an angle of 30°

$\displaystyle\large\underline{\sf\blue{To \ Find}}$

◈ The area of the shaded region?

$\displaystyle\large\underline{\sf\gray{Solution}}$

So the area of a sector is given by,

$\displaystyle\underline{\boxed{\sf Area \ of \ sector = \dfrac{\theta}{360^{\circ}} \times \pi r^2}}$

We shall subtact the area of the two sectors to find the area of the shaded region!!

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$\underline{\bigstar\:\textsf{According to the given Question :}}$

Area of a sector of the circle with radius 3.5 cm

➝ $\displaystyle\sf Area \ of \ sector = \dfrac{\theta}{360^{\circ}} \times \pi r^2$

➝ $\displaystyle\sf \dfrac{30^{\circ}}{360^{\circ}} \times \dfrac{22}{7}\times 3.5^2$

➝ $\displaystyle\sf \dfrac{1}{12}\times \dfrac{22}{7} \times 12.25$

➝ $\displaystyle\sf \dfrac{2}{7} \times 12.25$

➝ $\displaystyle\sf \purple{Area \ of \ sector_1 = 3.5 \ cm^2}$

Area of a sector of the circle with radius 7 cm

»» $\displaystyle\sf Area \ of \ sector = \dfrac{\theta}{360^{\circ}} \times \pi r^2$

»» $\displaystyle\sf \dfrac{30^{\circ}}{360^{\circ}} \times \dfrac{22}{7}\times 7^2$

»» $\displaystyle\sf \dfrac{1}{12}\times \dfrac{22}{7} \times 49$

[12 & 22 gets cancelled = 2 and also 7 & 49 gets cancelled = 7]

»» $\displaystyle\sf 2\times 7$

»» $\displaystyle\sf \orange{Area \ of \ sector_2 = 14 \ cm}$

Now the area of shaded region will be,

➳ $\displaystyle\sf Area \ of \ sector_2 - Area \ of \ sector_1$

➳ $\displaystyle\sf 14-3.5$

➳ $\displaystyle\sf \pink{10.5 cm^2}$

$\displaystyle\sf \therefore\:\underline{\sf Area \ of \ the \ shaded \ region \ is \ 10.5cm^2}$

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