geometrytriangles
Pythagoras shows us for a right angled triangle how to find the remaining side length where the lengths connected by 90 degrees are known.
But more-so there is a surprisingly short equation that I can't find that gives the remaining length where any two lengths are known. What is this equation?
Are there other equations relating the internal angles of two corners and the third, if so what are they?
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.
solution sketch:
Note that $h$ divides the triangle into two right triangles (h is the perpendicular bisector (altitude) from the base to the opposite vertex); the angles line $h$ forms with $a$ are right angles. This gives you two right triangles, and you only need one of them to compute the values you need.
Call the two known (marked) angles $\theta$ (they are equal).
If you know the length of the side $b$: $\sin\theta = \dfrac{h}{b} \implies h = b\sin\theta$
If $a$ is known, $\tan\theta = \dfrac{h}{a/2} \implies h = \dfrac{a}{2}\tan \theta$.
Now, using the pythorean theorem to relate your sides, we know that
$$h^2+\left(\frac{a}{2}\right)^2\,=\;b^2\tag{pythogorean theorem}$$
If the length of $b$ is unknown, using the pythagorean theorem, then knowing $a/2$ and $h$ will allow you to solve for $b$.
Knowing $\,b\,$ and $\,h\,$ will allow you to solve for $\,\dfrac{a}{2}\,$ by the pythagorean theorem Then double the value of $a/2$ to get $a$.
Best Answer
When you know all angles and two of the sides, you can choose between two different rules for finding the third side. Both apply to general triangles:
The law of cosines: $a^2+b^2-2ab\cos C=c^2$. This can be thought of as a generalization of Pythagoras' theorem because the cosine term vanishes when $C$ is a right angle.
The law of sines: $\frac{\sin C}{c}=\frac{\sin A}{a}$. This can be solved for $c$ to get $c=a\frac{\sin C}{\sin A}$.
If you google for solving triangles you will find numerous walk-throughs of how to find the entire triangle depending on what you already know about it. The first hit for me is here, which looks good.
In general you need three pieces of information, where a "piece" is either the length of a side or an angle. Three pieces also turn out to be sufficient except in the AAA case (you know all angles and no length, so any solution can be scaled by an arbitrary abount and still be a solution) and the SSA case (where you know to sides and one of the angles, but not the angle where the known sides meet; then there can be two different solutions).