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15. A regular hexagon is having side of 12 cm. Find the area of the hexagon. 

16.. In a trapezium. one of the parallel sides is half of the other. The height is 4 cm. If the area of the trapezium is 36, then what are the lengths of the parallel sides? 

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15. Given it is a regular hexagon with side 12 cm.
​This regular word shows that all the sides of the hexagon are same ie. 12 cm.
​Now if we divide a regular hexagon it gives us 6 equilateral triangles of all sides 12 cm.

​As we all know area of equilateral triangles without height = (Root 3/4) aunit​2
​Where a = Side of the triangle (Equilateral)

By putting the values we get = (Root 3/4) 122
​So the answer will be = 0.433...*144
Area = 62.3538 cm​2

​At the start we got 6 equilateral triangles so we have to multiply 6 to the value of Area of one equilateral triangle
​ie. 62.3538 cm* 6
Thus Area of Hexagon = 374.1228 cm2

16. Here Height = 4 cm , Parrallel sides => 1st=x ,2nd = x/2 And Area = 36 cm2
​Area of Trapezium = 1/2 *(Sum of parrallel sides) * height

​​So by putting the values to the above formula, it gives us
(x+x/2)/2 * 4 = 36 cm​2    ( By shifting 4 to other side of equal to RHS)
​(x+x/2)/2 = 36/4 ==> 1/2 * (x+x/2) = 9*2  (By shifting 2 to other side RHS)
​(2x+x)/2=18 ==> (By taking 2 to other side RHS)
3x=18*2 ==> 3x=36 (By taking 3 to RHS)

​Put the value of x to the given parrallel sides in the question
ie x and x/2
​Thus, The parrallel sides will be 12cm and 6 cm
For proof use these values in the formula ie 1/2 * (12+6) * 4
Area of Trapezium = 9*4 = 36cm2 Hence proved                                                      
​Hence, The 2 parrallel sides are 6cm and 12 cm respectively.
I hope it has helped you...

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