12 Sided Polygon

12 Sided Polygon A geometric shape which is enclosed by 12 lines is known as 12 sided polygon, it is also known asdodecagon. The area of a regular polygon: The area of a regular polygon of n sides = (na^2/4) cot( /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 hexagonthen it means n= 6, now we can substitute n= 6 in equation (1) =(6a^2 / 4) cot ( / 6) =(33 / 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( /n) = (12 a^2 / 4) cot ( / 12). Now we need to find the value of cot ( / 12). Suppose a/2 = / 12 a = 2( /12) = /6 Since cot (a/2) = (1 + cos a) / sin a Therefore, cot ( /12) = [1 + cos ( /6)]/ sin ( / 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)