[Math] How to determine the side lengths of an irregular polygon when all interior angles are known

geometry

  1. Given an irregular polygon where all of the angles are known, how many side lengths need to be known, at minimum, to determine the length of the remaining sides?

  2. Given all the angles and the requisite number of side lengths, how to actually calculate the remaining side length?

Example: a 5 Sided polygon's interior angles will add up to 540 degrees. ((5-2)*180=540).

Given the following interior angles:

AB 140 degrees
BC 144 degrees
CD 78 degrees
DE 102 degrees
EA 76 degrees

And knowing that Side A is 12 units long, can we determine the remaining side lengths? Or are more side lengths needed?

UPDATE:

Since you need three consecutive side lengths of a five sided figure, I'm adding three sides here so I can see an example of how the calculations are done for the remaining two sides:

Side A = 27 7/8"
Side B = 7"
Side c = 13 1/4"

Best Answer

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.

A (very crude) pentagon

Related Question