# In the figure given below find the area of the shaded region where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as center.

Last updated date: 18th Mar 2023

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Answer

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Hint: - Area of shaded region $ = $Area of circle$ + $Area of equilateral triangle$ - $area of

common region.

Given data:

Radius of circle$\left( r \right) = 6m$

Side of an equilateral triangle$\left( a \right) = 12cm$

As we know area of circle$ = \pi {r^2} = \dfrac{{22}}{7} \times {6^2} = \dfrac{{792}}{7}c{m^2}$

Now as we know area of equilateral triangle$ = \dfrac{{\sqrt 3 }}{4}{a^2} = \dfrac{{\sqrt 3 }}{4} \times

{12^2} = 36\sqrt 3 c{m^2}$

Area of common region (i.e. between circle and equilateral triangle)

$ \Rightarrow \left( {\dfrac{\theta }{{{{360}^0}}}} \right)\pi {r^2}$

As we know in equilateral triangles all angles equal to${60^0}$.

\[ \Rightarrow \angle {\text{AOB}} = {60^0} = \theta \]

Therefore area of common region$ = \left( {\dfrac{\theta }{{{{360}^0}}}} \right)\pi {r^2} =

\dfrac{{{{60}^0}}}{{{{360}^0}}} \times \dfrac{{22}}{7} \times {6^2} = \dfrac{{132}}{7}c{m^2}$

Therefore, the area of the shaded region$\left( A \right)$$ = $Area of circle$ + $Area of equilateral

triangle$ - $area of the common region.

$ \Rightarrow \left( A \right) = \dfrac{{792}}{7} + 36\sqrt 3 - \dfrac{{132}}{7} = \left( {\dfrac{{660}}{7} +

36\sqrt 3 } \right)c{m^2}$

So, this is the required answer.

Note: -In such types of questions always remember the formula of area of standard shapes which is

stated above, then first find out the area of circle then find out the area of triangle then find out the

area of common region, then find out the area of shaded region using the formula which is stated above

then simplify we will get the required answer.

common region.

Given data:

Radius of circle$\left( r \right) = 6m$

Side of an equilateral triangle$\left( a \right) = 12cm$

As we know area of circle$ = \pi {r^2} = \dfrac{{22}}{7} \times {6^2} = \dfrac{{792}}{7}c{m^2}$

Now as we know area of equilateral triangle$ = \dfrac{{\sqrt 3 }}{4}{a^2} = \dfrac{{\sqrt 3 }}{4} \times

{12^2} = 36\sqrt 3 c{m^2}$

Area of common region (i.e. between circle and equilateral triangle)

$ \Rightarrow \left( {\dfrac{\theta }{{{{360}^0}}}} \right)\pi {r^2}$

As we know in equilateral triangles all angles equal to${60^0}$.

\[ \Rightarrow \angle {\text{AOB}} = {60^0} = \theta \]

Therefore area of common region$ = \left( {\dfrac{\theta }{{{{360}^0}}}} \right)\pi {r^2} =

\dfrac{{{{60}^0}}}{{{{360}^0}}} \times \dfrac{{22}}{7} \times {6^2} = \dfrac{{132}}{7}c{m^2}$

Therefore, the area of the shaded region$\left( A \right)$$ = $Area of circle$ + $Area of equilateral

triangle$ - $area of the common region.

$ \Rightarrow \left( A \right) = \dfrac{{792}}{7} + 36\sqrt 3 - \dfrac{{132}}{7} = \left( {\dfrac{{660}}{7} +

36\sqrt 3 } \right)c{m^2}$

So, this is the required answer.

Note: -In such types of questions always remember the formula of area of standard shapes which is

stated above, then first find out the area of circle then find out the area of triangle then find out the

area of common region, then find out the area of shaded region using the formula which is stated above

then simplify we will get the required answer.

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