Is there a variation of Heron's Formula for pentagons, hexagons, etc? Prof. Loh showed us one for quadrilaterals.
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Module 2 Week 4 Day 16 Bonus Explanation Part 1
Title says it all!
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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.