Question

What is the formula to find out the area of irregular pentagon?

Answer 1

There is no such formula.
However, with some more information known about this pentagon, the area can be determined. See below.

Explanation:

There can be no such formula because a pentagon is not a rigid polygon. Given all its sides, the shape is still not defined and, therefore, the area cannot be determined.

However, if you can inscribe a circle into this pentagon and know its sides an a radius of the inscribed circle, the area can easily be found as
`S = (p*r)/2`
where `p` is a perimeter (sum of all sides) and `r` is a radius of inscribed circle.

Proof of the above formula is easy. Just connect a center of an inscribed circle with all vertices and consider all triangles formed by this construction. Their bases are sides of the pentagon and each of their altitudes is a radius of an inscribed circle.