# Contest problem 5

• Mar 15th 2012, 11:11 AM
Kanwar245
Contest problem 5
The game Hexagon consists of some pieces, each in a shape of a regular hexagon with the area 6. Each piece is divided by lines into four right triangles, six isosceles triangles and one quadrilateral (see the picture). What is the area of the shaded quadrilateral?
• Mar 15th 2012, 11:47 AM
princeps
Re: Contest problem 5
Quote:

Originally Posted by Kanwar245
The game Hexagon consists of some pieces, each in a shape of a regular hexagon with the area 6. Each piece is divided by lines into four right triangles, six isosceles triangles and one quadrilateral (see the picture). What is the area of the shaded quadrilateral?

$A_H=\frac{3}{2}a^2 \sqrt 3 \Rightarrow 6=\frac{3}{2}a^2 \sqrt 3 \Rightarrow a^2 \sqrt 3=4 \Rightarrow a^2=\frac{4}{\sqrt 3}$

$A_Q=2\left(\frac{a\cdot a \sqrt 3}{8}\right)=\frac{a^2 \sqrt 3}{4}=1$