AOB is a quadrant of a circle of radius 14 cm. from it an equilateral triangle OPQ is cut off as shown in the figure calculate the area of the shaded part

Dear Student,

Please find below the solution to the asked query:

Given : AOB is quadrant of circle ,

We know area of quadrant of circle  = $\frac{\mathrm{\pi } r^2}{4}$ , here r  = 14 cm  (  As given radius  =  14 cm ) , So

Area of quadrant AOB  = $\frac{{\displaystyle \frac{22}{7}\times 14 \times 14}}{4} = \frac{{\displaystyle 22 \times 2 \times 14}}{4}= \frac{{\displaystyle 22 \times 14}}{2}= 22 \times 7$ = 154 cm²

Also given : OPQ is a equilateral triangle and we can see from the given diagram OP and OQ are radius of quadrant AOB , So 

Side of equilateral triangle OPQ  =  14 cm

We know area of equilateral triangle  = $\frac{\sqrt{3}}{4}{\left(\mathrm{Side}\right)}^2$ ,Here Side = 14 cm , So

Area of equilateral triangle OPQ  = $\frac{\sqrt{3}}{4}{\left(14\right)}^2 = \frac{\sqrt{3}}{4}\times 14 \times 14= \sqrt{3}\times 7 \times 7 = 1.732\times 7 \times 7$ = 84.868 cm²

And

Area of shaded region  =  Area of quadrant AOB  -  Area of equilateral triangle OPQ

Therefore,

Area of shaded region  =  154 - 84.868 = 69.132 cm²                                                   ( Ans )


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