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.