A rectangle is formed inside a regular hexagon. If the diagonal of the rectangle is 10 find the area of the hexagon

Question

A rectangle is formed inside a regular hexagon. If the diagonal of the rectangle is 10 find the area of the hexagon

Answer 1

A very simple formula to solve for the area of any regular polygon is

A= (.25)(√3)(s2)(# of sides)

This formula is derived from the fact that all regular polygons are made of equilateral triangles, but I won’t go into that here.

Anyway, now we need to find the number of sides and the length of the sides. We know that we are dealing with a hexagon which has 6 sides so we can update or formula to

A= (.25)(√3)(s2)(6)

Ok so the problem says that the diagonal of the rectangle is 10. The diagonal of the rectangle is a similar concept to the diameter of a circle. So just like half of the diameter gives you the radius of a circle, half of the diagonal gives you the radius of the hexagon. In a regular polygon the radius is equal to the side. So half of 10 is 5, so we now know that the side length equals 5. So let’s update our formula to

A= (.25)(√3)(52)(6)

A= (.25)(√3)(25)(6)

A= 37.5√3

A= 64.95190528383289850727923780647

Which of course can be rounded to

A= 65

I hope this helps :-)

Answer 2

You might want to find the exact solution. Here you go:

area of the hexagon = (5+10)[(5/2)√3] = (75/2)√3 unit^2

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Ideas: Double a trapezoid area. 2(1/2)(base 1 + base 2)(height) = (base 1 + base 2)(height), while the height can be found by 30-60-90 degree special triangle.

 

 

Answer 3

Hey Sarae -- a hex is made of 6 equilateral triangles ... a rect diagonal of 10 means a tri side of 5 ... let the tri base be 5 and the height is about 86% of the base or 4.3 ... 6 tri areas is 3bh= 3(5)4.3= 13(5)= about 65 Regards :)