# Ex.12.2 Q14 Areas Related to Circles Solution - NCERT Maths Class 10

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## Question

Tick the correct answer in the following:

Area of a sector of angle $$p$$ (in degrees) of a circle with radius $${R}$$ is

(A) \begin{align}\frac{{P}}{180^{\circ}} \times 2 \pi {R}\end{align}

(B) \begin{align}\frac{{P}}{180^{\circ}} \times 2 \pi {R}^{2} \end{align}

(C) \begin{align}\frac{{P}}{720^{\circ}} \times 2 \pi {R}\end{align}

(D) \begin{align}\frac{{P}}{720^{\circ}} \times 2 \pi {R}^{2}\end{align}

## Text Solution

What is known?

A sector of angle $$p$$ (in degree) of a circle with radius $${R.}$$

What is unknown?

Area of a sector.

Reasoning:

Consider

Area of the sector of angle \begin{align}\theta =\frac{{ }\!\!\theta\!\!{ }}{{360}}{ }\!\!\times\!\!{ }\,\pi {{{r}}^{{2}}}\end{align}

where $${r}$$ is the radius of the circle

Here $${θ = p}$$ and $${r = R}$$

$$\therefore$$ Substituting above values in formula we get

Area of the sector \begin{align} = \frac{{{p}}}{{{{36}}{{{0}}^{{o}}}}}{\times \pi }{{{R}}^{{2}}}\end{align}

Multiplying numerator and denominator of formulas obtained above  by 2 we get

Area of the sector \begin{align} = \frac{{{p}}}{{{{72}}{{{0}}^{{o}}}}}{\times 2\pi }{{{R}}^{{2}}}\end{align}

Steps:

If radius of a circle $$= {R}$$

We know, Area of sector of angle \begin{align}= \,\frac{\theta }{{{{36}}{{{0}}^{{o}}}}}{ \times \pi }{{{R}}^{{2}}}\end{align}

$$\therefore$$ Area of sector of angle $$p$$

\begin{align} &= \frac{{{p}}}{{{{36}}{{{0}}^{{o}}}}}{ \times \pi }{{{R}}^{{2}}}\\ \,\,\,\,\,\,\,\,\,& = \frac{{{2}}}{{{2}}}\left( {\frac{{{p}}}{{{{36}}{{{0}}^{{o}}}}}{ \times \pi }{{{R}}^{{2}}}} \right)\\\, \,\,\,\,\,\,\,\,\, &= \frac{{{p}}}{{{{72}}{{{0}}^{{o}}}}}{ \times 2\pi }{{{R}}^{{2}}}\end{align}

Hence D is the correct answer.

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