Question

In figure, are shown two arcs $P A Q$ and $P B Q$ . Arc $P A Q$ is a part of circle with centre $O$ and radius $O P$ while arc $P B Q$ is a semi circle drawn on $P Q$ as diameter with centre $M$ . If $O P = O Q = 10 cm$ , show the area of shaded region is $25 (\sqrt{3} - \frac{π}{6}) {cm}^{2}$ .

Accepted Answer

Given: $O P = O Q = 10 cm$ .

$PQ = 10 cm$ .

So, $P Q$ is an equilateral triangle.

$∠ P O Q = 60 °$

Area of an equilateral triangle $25 (\sqrt{3} - \frac{π}{6}) {cm}^{2}$

We know that: Area of sector $= \frac{θ}{360} \times π r^{2}$ ${cm}^{2}$

Area of sector $O P A Q =$ $\frac{60}{360} \times {πr}^{2} = \frac{100 π}{6} {cm}^{2}$

We know that area of the semicircle $= \frac{1}{2} \times {πr}^{2}$ ${cm}^{2}$

Area of the region $P A Q M =$ Area of sector $O P A Q -$ Area of equilateral triangle $O P Q$ $= \frac{100 π}{6} - 100 \frac{\sqrt{3}}{4} = 100 (\frac{π}{6} - \frac{\sqrt{3}}{4})$ ${cm}^{2}$

Area of the shaded region $=$ Area of semicircle $P Q B$ $-$ Area of the region $P A Q M$ $= \frac{1}{2} \times 25 \times π - 100 (\frac{π}{6} - \frac{\sqrt{3}}{4})$ ${cm}^{2}$

Area of the shaded region $= 25 \sqrt{3} - 25 \frac{π}{6} = 25 (\sqrt{3} - \frac{π}{6}) {cm}^{2}$

Hence, the area of the shaded region $= 25 (\sqrt{3} - \frac{π}{6}) {cm}^{2}$ .

Marks	$20 - 30$	$30 - 40$	$40 - 50$	$50 - 60$	$60 - 70$	$70 - 80$	$80 - 90$
Frequency	$p$	$15$	$25$	$20$	$q$	$8$	$10$

In figure, are shown two arcs $P A Q$ and $P B Q$ . Arc $P A Q$ is a part of circle with centre $O$ and radius $O P$ while arc $P B Q$ is a semi circle drawn on $P Q$ as diameter with centre $M$ . If $O P = O Q = 10 cm$ , show the area of shaded region is $25 (\sqrt{3} - \frac{π}{6}) {cm}^{2}$ .

Important Questions on Model Test Paper

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In figure, are shown two arcs PAQ and PBQ. Arc PAQ is a part of circle with centre O and radius OP while arc PBQ is a semi circle drawn on PQ as diameter with centre M. If OP=OQ=10 cm, show the area of shaded region is 253-π6 cm2.

Important Questions on Model Test Paper

Learn and Prepare for any exam you want !

In figure, are shown two arcs $P A Q$ and $P B Q$ . Arc $P A Q$ is a part of circle with centre $O$ and radius $O P$ while arc $P B Q$ is a semi circle drawn on $P Q$ as diameter with centre $M$ . If $O P = O Q = 10 cm$ , show the area of shaded region is $25 (\sqrt{3} - \frac{π}{6}) {cm}^{2}$ .