So I took a challenge from my Geometry teacher to create code that when the user gives the computer how many angles / sides a polygon has and the angle of each of the interior angles it could find each of the exterior angles whether it is regular or irregular. For example the user tells the computer they have a four-sided shape (quadrilateral), the interior angles are $70, 75, 110, 145$. The computer would then be able to output each corresponding angle in the proper order. My problem is that with coding you can't just type in the human logic so I need a formula to work from: what on earth would be a formula that could apply to any polygon with any number of sides (irregular or regular)?
[Math] A formula for the exterior angles of an irregular polygon, given interior angles
algebra-precalculusgeometry
Related Solutions
The lengths don't determine the angles: for example with all sides equal you have a rhombus, where the angle at any vertex could be anything in the interval $(0, \pi)$.
For a $n$ sided polygon, you need all the angles in order and $n-2$ consecutive side lengths in order to construct the polygon. So, you need the lengths of sides $B,C$ or $E,B$ or $D,E$ to construct your polygon.
The best way to find out the length of the remaining side is by drawing diagonals and applying triangle laws (sine or cosine rule).
Consider the (very badly drawn) pentagon. It is not drawn to scale, but you get the idea. Here are the steps you will take to find out the lengths of $D,E$.
1.Find out length of $X$ using cosine rule in $\Delta ABX$.
2.Knowing $\angle a,X,A,B$, find out $\angle e,\angle b$ using sine rule.
3.$\angle c = \angle\mbox{(between B,C)} -\angle b$. So, $\angle c$ is known.
4.Repeat the whole procedure for $\Delta CXY$. Find out $Y,\angle d, \angle f$.
5.$\angle g, \angle h$ are easily calculated now.
6.$\angle i$ is known. Apply sine rule in $\Delta DEY$ to find out $D,E$-the two unknown sides.
Best Answer
At each vertex of the polygon, the interior angle and exterior angle must sum to $180$. You can use this to find the value of each exterior angle.