Question

Find the area of the shaded region in the given figure, where a circular arc of radius 7cm has been drawn with a vertex A of an equilateral triangle ABC of side 14 cm as centre.(Use$\pi  = \dfrac{{22}}{7}$ and $\sqrt 3  = 1.73$).

Answer 1

Hint: In questions, to solve areas of shaded figures we need to divide the question into different figures and find their respective areas and then calculate the area of the shaded figure by adding or subtracting their respective areas.

Complete Step-by-Step solution:

Given,

Radius of circle =7cm

$  \dfrac{1}{2} \times {r^2} \times \Phi  $

$   = \dfrac{1}{2} \times {(7)^2} \times (60 \times \dfrac{\pi }{{180}}) $

$   = 25.6c{m^2} $

In $\Delta ABC$ Side of each side = 14 cm

Area of shaded figure = area of circle + area of $\Delta ABC$- area of circular arc AED

Area of circle=$\pi {r^2}$

$ = (\dfrac{{22}}{7}) \times {(7)^2} $

$   = 153.86c{m^2} $

Area of $\Delta ABC$=$\dfrac{{\sqrt 3 }}{4} \times {(Side)^2}$

$ = \dfrac{{\sqrt 3 }}{4} \times {(14)^2}$

$ = 84.77c{m^2}$

Area of circular arc AED=$\dfrac{1}{2} \times {r^2} \times \theta $                                                                      …..($\theta $ is in terms of radians)

$   = \dfrac{1}{2} \times {(7)^2} \times (60 \times \dfrac{\pi }{{180}}) $

$   = 25.6c{m^2} $

Area of shaded figure = area of circle + area of $\Delta ABC$- area of circular arc AED

$=153.86+84.77-25.6$

$=212.78 c{m^2}$

Hence, the answer to this question is 212.78$c{m^2}$.

Note:  In this type of question, we need to carefully analyze the figures whose areas are needed to be calculated in order to calculate the question. Moreover, we need to remember the basic formulae of areas of figures.