The Area  Of  The  Pentagon

Figure 1

The area of the pentagon is the combined area of the 5 triangles shown in Figure 1, each of which has a vertex at O.
What is the area of each of these triangles?
Lets work with triangle ODE.
From   Pentagon Construction we know that the distance O to any vertex is
In order to get the area of a triangle we need to know OF.
DE = side of pentagon = s, so FE =  
height of triangle = OF² =  OE² - FE² =    =    

Now we can get the area of each triangle, for the area = 1/2 * base * height:

Area of pentagon = 5 * area of each triangle.

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