Remember, this is only for REGULAR PENTAGONS!
A regular pentagon can be divided into five congruent triangles.
Call the side length of the pentagon s.
Then s represents the base of each individual triangle.
To find the altitude (or apothem), you have to use trigonometry.
Take the base angle of one of these isosceles triangles.
This is just the pentagon interior angle (108 degrees) divided by two, or 54 degrees.
Because we are comparing the altitude (or apothem) to the base, using tangent (opposite/adjacent) is most appropriate.
To clarify that, divide the isosceles triangle into two right triangles with the apothem as a side.
Call the apothem length a.
We know the larger acute angle in the right triangle is 54 degrees. Its opposite side is the apothem, and its adjacent side is half of s (because the base was split into two equal sides).
Therefore, tan(54) is just (a)/(s/2).
Because we want to find (a) in terms of s, we can multiply tan(54) by (s/2) to cancel out the denominator of the ratio above.
An approximation of tan(54) is 1.376, so a = 1.376(s/2).
This simplifies to a = 0.6882(s).
Then the area of the whole isosceles triangle is s(0.6882(s))/2, or 0.3441(s^2).
Finally, multiplying the area by five will result in the area of the pentagon, which is 1.7205(s^2).