Question

OACB is a quadrant of a circle of radius 3.5cm with centre O. If OD =2cm, calculate the area of the shaded region. ​

Answer 1

Given : OACB is a quadrant of a circle of radius 3.5cm with centre O and OD = 2cm

To Find : Area of the shaded region.

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Firstly, we'll find the area of the quadrant OABC.

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• Radius = 3.5cm

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We know that :

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$\underline{\boxed{\bold\purple{\bigstar \: area \: of \: the \: quadrant \: = \frac{1}{4} \times area \: of \: circle}}}$

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Put the values in the formula -

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$\sf:\implies \: area \: of \: the \: quadrant \: oacb = \dfrac{1}{4} \bold{\pi}r {}^{2} \\ \\ \\ \sf:\implies \: area \: of \: the \: quadrant \: oacb = \frac{1}{4} \times \frac{22}{7} \times (3.5) {}^{2} \\ \\ \\ \sf:\implies \: area \: of \: the \: quadrant \: oacb = \frac{1}{4} \times \frac{22}{7} \times 3.5 \times 3.5 \\ \\ \\ \sf:\implies \: area \: of \: the \: quadrant \: oacb = \underline{\boxed{\bold\purple{9.625 {cm}^{2} }}}\bigstar$

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We'll find the area of the ∆AOD

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We know that :

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$\underline{\boxed{\bold\purple{\bigstar \: area \: of \: a \: traingle = \frac{1}{2} \times b \times h }}}$

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Where

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• b denotes the base of the triangle.
• h denotes the height of the triangle.

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Since, OABC is a quadrant

So, ∠AOD = 90°

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Now,

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$\sf:\implies \:area \: of \: the \: triangle \: = \dfrac{1}{2} \times b \times h \\ \\ \\ \sf:\implies \: area \: of \: the \: triangle \: = \frac{1}{2} \times 3.5 \times 2 \\ \\ \\ \sf:\implies \:area \: of \: the \: triangle = \frac{1}{\cancel{2}} \times 3.5 \times \cancel2 \\ \\ \\ \sf:\implies \: area \: of \: the \: triangle \: = \underline{\boxed{\bold\purple{3.5 {cm}^{2} }}}\bigstar$

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Now, we'll find the area of the shaded region.

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$\underline{\boxed{\bold\purple{\bigstar \: area \: of \: the \: shaded \: region \: = area \: of \: the \: quadrant - area \: of \: the \: triangle}}}$

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$\sf:\implies area \: of \: the \: shaded \: region = area \: of \: the \: quadrant - area \: of \: the \: triangle \\ \\ \\ \sf:\implies \: area \: of \: the \: shaded \: region = 9.625 - 3.5 \\ \\ \\ \sf:\implies \: area \: of \: shaded \: region = \underline{\boxed{\bold\purple{6.125}}}\bigstar \\ \\ \\ \\ \sf\therefore{\underline{Hence, \: the \: area \: of \: the \: shaded \: region \: is \: \bold{6.125cm} {}^{2} .}}$