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
•
Total views: 307.5k
•
Views today: 3.86k
Answer
307.5k+ views
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.
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
