The length of each side of a regular hexagon measures . Find the area of the inscribed regular hexagram shaded in the figure.
The area of the regular hexagram.
The length of each side of a regular hexagon is .
Consider the regular hexagon,
Area of the regular hexagon is where is the side length of the hexagon.
given each side of a hexagon is . i.e.,
Draw a line such that is perpendicular to and is the midpoint of
Since is a right angle triangle and the angle of the triangle is .
Therefore by the properties of triangle the hypotenuse is equal to twice the length of the shorter leg, which is the side across from the angle. The longer leg, which is across from the angle, is equal to multiplying the shorter leg by the .
Substitute in , we get
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