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What is the area enclosed between the circles? Given that each circle is having a radius of $2 cm .$

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4 \sqrt{3} c m^{2}

(2 \sqrt{3} - π) c m^{2}

2 \sqrt{3} - 2 π) c m^{2}

2 (2 \sqrt{3} - π) c m^{2}

answer is D.

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Detailed Solution

Given,

The radii of the circles with centre

A

and

B

is equal to

r = 2 cm .

Since they touch each other

AB

will pass through the point of contact of those circles and,

AB = 2 r

\Rightarrow

= 2 \times 2

\Rightarrow

= 4 cm

Similarly,

BC = 4 cm

AC = 4 cm

△ ABC

is an equilateral triangle.
So,

ar (△ ABC) = \frac{\sqrt{3}}{4} a^{2}

\Rightarrow

ar (△ ABC) = \frac{\sqrt{3}}{4} \times 4 \times 4

\Rightarrow

ar (△ ABC) = 4 \sqrt{3} c m^{2}

△ ABC

crops

3

sectors with each of them having radius equal to

2 cm

and central angle equal to respective vertex angle of the

△ ABC

that is equal to

60^{°} .

So, the three sectors are equal in area.
Area of a sector is

\frac{θ^{°}}{360^{°}} \times π r^{2}

.
Sum of area of

3

sectors

=

3

\times

Area of

1

sector.

\Rightarrow

Sum of area of

3

sectors

= 3 \times \frac{θ^{°}}{360^{°}} \times π r^{2}

\Rightarrow

Sum of area of

3

sectors

= 3 \times \frac{60^{°}}{360^{°}} \times π \times 2^{2}

\Rightarrow

Sum of area of

3

sectors

= 2 π {cm}^{2}

Now, area between circles

=

Area of

△ ABC

-

Area of

3

sectors.

\Rightarrow

Area between circles

= 4 \sqrt{3} - 2 π

\Rightarrow

Area between circles

= 2 (2 \sqrt{3} - π) {cm}^{2}

Hence, option 4 is correct.

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What is the area enclosed between the circles? Given that each circle is having a radius of 2 cm.

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What is the area enclosed between the circles? Given that each circle is having a radius of $2 cm .$