THE AREA OF A PENTAGRAM


AND REGULAR POLYGONS


AREA OF PENTAGRAM

There is more than one way to find the area of a pentagram. The method we will use here is to work with the areas of isosceles triangles. In the figure to the left the area of the regular pentagon can be found by multiplying the area of the yellow isosceles triangle by 5. This includes the pentagram along with 5 isosceles triangles each equivalent to the red isosceles triangle. Therefore, we will subtract from the area of the pentagon 5 times the area of the red isosceles triangle, leaving the area of the pentagram.


The area of any isosceles triangle is given by the formula:

A = s2Tan( a )/4


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 s2Ö3/4 (a familiar result).

In our case that angle is 54 degrees or 3p/10 radians. Hence, the area of the regular pentagon is

Pentagon = 5A = 5s2Tan( 3p/10 )/4


Now for the red isosceles triangle the base angle is 36 degrees or p/5 radians.
So the area of that triangle is s2Tan( p/5 )/4.
Hence, the area of the pentagram is

Pentagram = 5s2Tan( 3p/10 )/4 - 5s2Tan( p/5 )/4
             = (5s2/4)(Tan( 3p/10 ) - Tan( p/5 ))


Here are some problems for you to try:


1. Circumscribe a circle about a square (see the sketch)
    and use the method above, with the
    appropriate formulas, to show that the area
    of the square is s2, as expected.

 AREA OF SQUARE


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.



4. Circumscribe a circle about a regular n-gon (see the sketch)
    and use the method above to find the area
   of the n-gon in terms of n and s.

AREA OF n-GON
5. Find another method to find the area of the pentagram.

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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             © Thomas M. Green, Contra Costa College