Brahmagupta's formula, Heron's formula, and the circumscription of polygons

I noticed recently that the formulas of Heron and Brahmagupta have the same structure, and thus I conjecture that any polygon with sides (a, b, c, d, etc.) and a semi-perimeter (s) that can be circumscribed will have an area (A) given by the following equations:

s = (a + b + c + d ....)/2

A = sqrt[(s - a)(s -b)(s - c)(s - d)....]

Heron's formula (for the area of any triangle)

s = (a + b + c)/2

A = sqrt[(s - a)(s -b)(s - c)]

Brahmagupta's formula (for any cyclic quadrilateral)

s = (a + b + c + d )/2

A = sqrt[(s - a)(s -b)(s - c)(s - d)]