A regular hexagon has side 2 meters and has a circumscribed circle. What is the area of the hexagon, and what is the area of the circumscribed circle?

1 Answer
Jan 4, 2016

The area of hexagon #=color(blue)(10.932 m^2#

area of the circle : #color(blue)(=12.56 m^2#

Explanation:

http://etc.usf.edu/clipart/43400/43447/6c2_43447.htm

1) Hexagon
The formula for area of a hexagon is :
#color(blue)(area =(3sqrt3)/2 xx (side)^2#

The side #=2m#

#area =(3sqrt3)/2 xx (2)^2#

#=(3sqrt3)/cancel2 xxcancel4#

#=3sqrt3 xx2#

#=6sqrt3#
(#sqrt3 = 1.732)#

#=6xx1.732#

The area of hexagon #=color(blue)(10.932 m^2#

2) Circle

Observe the above diagram , each of the triangles within the hexagon are equilateral.

The angle around a point #=360^o#

So, the angle of each triangle #=360/6=60^o#

Now observe a triangle , #2# sides are radii of the circle itself and as per property : angles opposite equal sides are equal, so all angles of the triangle are# =60^o#.
Thus , the triangles are all equilateral .

The side of the hexagon #=# radii of the circle.

#r= 2m#

So, the area #=pi xx (r)^2#

#=3.14 xx 2^2#

#=3.14 xx 4#
area of the circle : #color(blue)(=12.56 m^2#