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The area of a regular polygon of n sides =

Where a is the length of a side of the polygon. In particular, if a be the length of a side of a regular hexagon then it means n= 6, now we can substitute n= 6 in equation (1)

=(6a^2 / 4) cot (π / 6)

=(3√3 / 2) a^2

Therefor n = 12

The area of a regular polygon of n sides= (na^2/4) cot(π /n)

= (12 a^2 / 4) cot (π / 12).

Now we need to find the value of cot (π / 12).

Suppose a/2 = π / 12

a = 2(π /12) = π /6

Since cot (a/2) = (1 + cos a) / sin a

Therefore, cot (π /12) = [1 + cos (π /6)]/ sin (π / 6)

= (1 + √3/2) / (1/2)

= (2 + √3)

Therefore

Solution: - According to the problem a= 5 then

Area of dodecagon = 3 (2 + √3) a^2

= 3 (2 + √3) 5^2

= 75 (2 + √3)