Hi @mirthfulcygnet,
Nice to hear from you! It is great that you are continuing to be interested and ask questions even after having finished the lessons!
It is true that there're exists a generalizations of Heron's Formula for quadrilaterals and pentagons, but they are much-much more complicated and oriented for higher-level mathematicians. The same relatively easy formulas for the areas of polygons with more than three angles, analogous to Heron's Formula, do not exist.
To begin with, Heron's formula is a special case of Brahmagupta's formula for the area of a cyclic quadrilateral. Brahmagupta's formula, in its turn, is a special case of Bretschneider's formula for the area of any quadrilateral.
Heron's formula is also a special case of the formula for the area of a trapezoid based only on its sides. Heron's formula is obtained by setting the smaller parallel side to zero.
One of the even more high-level formulas is a formula for Robbins pentagons: a cyclic pentagons whose side lengths and area are all rational numbers. If you really want to, you can read about it here (page 15), but I think it would be better for you to learn more in this area first. 🙂
Try to get used to the Heron's formula first and be ready to use it any time you need to! Then you will be ready to learn more such formulas.
