Question

three one rupee coin of unit radius each are kept intact such that each of them touch other two externally. What is the area of gap region so formed ? What is your answer when 4 coins are used​

Answer 1

Answer:

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Step-by-step explanation:

Radius of each coin =1 cm

With all the three centres an equilateral triangle of side 2 cm is formed.

Area enclosed by coind = Area of equilateral triangle −3× Area of sector of angle 60  

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      =  2 3 (2)  2 3×  360 60 ×π(1)  2  

       = 4 3 ×4−3×  6 1 ×π

         =(  3  −  2 π  ) cm  2

Answer 2

Answer:

(√3-π/2) r^2

Step-by-step explanation:

Let the unit radius be r.

When the coins are placed as mentioned in the question, the distance between any two centres is 2r.

Thus, by joining the three centres we get an equilateral triangle of side 2r.

Each angle of an equilateral triangle measures 60°.

Area of Equilateral triangle

=√3/4*(side) ^2

=√3/4*(2r) ^2

=√3/4*4r^2

=√3r^2

Notice that there is 1 sector of each coin (3 in total) inside the equilateral triangle.

Since the 3 coins have unit radius and the angle of each sector is 60°(angles of an equilateral triangle), each sector is identical.

Area of each sector

=(Angle of the sector) /360° * πr^2

=60°/360° * πr^2

=1/6 * πr^2

Therefore, Area of Gap region=Area of Equilateral triangle-3(Area of each sector)

=(√3r^2)-3(π/6*r^2)

=r^2{√3-3*π/6)

=r^2{√3-π/2}

={√3-π/2}r^2

I hope this will help you...