What is the area of a regular hexagon circumscribed iinside a circle with a radius of 1?

1 Answer
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Dec 18, 2015

#frac{3sqrt{3}}{2}#

Explanation:

The regular hexagon can be cut into 6 pieces of equilateral triangles with length of 1 unit each.

For each triangle, you can compute the area using either

1) Heron's formula, #"Area"=sqrt{s(s-a)(s-b)(s-c)#, where #s=3/2# is half the perimeter of the triangle, and #a#, #b#, #c# are the length of the sides of the triangles (all 1 in this case). So #"Area"=sqrt{(3/2)(1/2)(1/2)(1/2)}=sqrt{3}/4#

2) Cutting the triangle in half and applying Pythagoras Theorem to determine the height (#sqrt{3}/2#), and then use #"Area"=1/2*"Base"*"Height"#

3) #"Area"=1/2 a b sinC=1/2 (1) (1) sin(pi/3)=sqrt{3}/4#.

The area of the hexagon is 6 times the area of the triangle which is #frac{3sqrt{3}}{2}#.

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