THE AREA OF A PENTAGRAMAND REGULAR POLYGONS  
 
The area of any isosceles triangle is given by the formula:
where s is the length of the base of the isosceles triangle (and, in our case, the side of the pentagon) and a is the angle formed by the base and one of the two equal sides of the isosceles triangle. Note: For an equilateral triangle a = p/3 (60 degrees) and Tan( p/3 ) = Ö3 and the area of the equilateral triangle is s^{2}Ö3/4 (a familiar result).
In our case that angle is 54 degrees or 3p/10 radians. Hence, the
area of the regular pentagon is
Now for the red isosceles triangle the base angle is 36 degrees
or p/5 radians. = (5s^{2}/4)(Tan( 3p/10 )  Tan( p/5 )) Here are some problems for you to try:
2. Circumscribe a circle about a regular hexagon (make a sketch) and use the method above to find the area of the hexagon. 3. Circumscribe a circle about a regular octagon (make a sketch) and use the method above to find the area of the octagon.
6. Use the fact that the ratio of the diagonal of a regular pentagon to the the side of a regular pentagon is the golden ratio to find the area of the pentagram.

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