Question

Three circles of radius 1 touch each other externally as shown. The shaded area of the gap as shown
is equal to:​

Answer 1

Answer:

$\frac{7\sqrt{3}-11 }{7}$cm²

Step-by-step explanation:

As we know that area of shaded region= Area of equilateral triangle of side 2cm- 3* area of sector of having radius 1 cm and 60°

=> $\frac{\sqrt{3} }{4}a^{2}- 3*\frac{60 }{360} \pi r^{2}$

here, a=2cm and r= 1 cm

=> $\frac{\sqrt{3} }{4}2^{2} - \frac{180}{360}*\frac{22}{7} *1$

=> $\frac{7\sqrt{3}-11 }{7} cm^{2}$

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Answer 2

Answer:

The answer is $\frac{ [7\sqrt{3} - (11)]}{7}$ units

Step-by-step explanation:

Given data radius of the circles r = 1 unit

If join the centers of the 3 circles an equilateral triangle will be formed

then each side of the triangle = 1+1 = 2 units

And the area of the shaded part = Area of the triangle - 3( area sector)  

Now we will find area of the equilateral triangle and area of the sector

Area of the equilateral triangle

As we know area of equilateral triangle = $\frac{\sqrt{3} }{4} a^{2}$  

where $a$ is side of the triangle  

Therefore, area of the triangle =$\frac{\sqrt{3} }{4} (2)^{2}$ = $\frac{\sqrt{3} }{4} (4)$ = $\sqrt{3}$ units  

Area of the sector

Area of the sector = (θ/360°) × πr²,      

Where θ is angle of sector which is formed at center  

As we know angle in a equilateral triangle is equals to 60°  

⇒ angle of sector θ = 60°  

Area of the sector = $\frac{60}{360} (\pi )(1^{2} )$ =  $\frac{1}{6} (\frac{22}{7} )$ = $(\frac{11}{21} )$ units

Area of the shaded part = $\sqrt{3} - 3(\frac{11}{21} )$

                                         =  $\frac{21\sqrt{3} -3 (11)}{21}$

                                         = $\frac{3[7\sqrt{3} - 11]}{21}$

                                         = $\frac{ [7\sqrt{3} - 11]}{7}$ units

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