12 Sided Polygon

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A geometric shape which is enclosed by 12 lines is known as 12 sided polygon, it is also known as dodecagon.
 
The area of a regular polygon:

The area of a regular polygon of n sides = (na^2/4) cot(π /n)            … (1)

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

 
Example: - What is the area of dodecagon.
 
Solution: - Since 12 sided polygon is known as dodecagon.

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 Area of a dodecagon = 3(2 + √3) a^2

 
Example 2: - If each sides of dodecagon are 5 cm then find the area of dodecagon.
 
Solution: - According to the problem a= 5 then

            Area of dodecagon = 3 (2 + √3) a^2

                                           = 3 (2 + √3) 5^2

                                           = 75 (2 + √3)
 

 
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