Finding angles in a four sided room

[Hemmer]

This is a little problem that I encountered while trying to measure the floor plan of an irregularly shaped reverberation chamber. I have measured the length of the four walls (all of different lengths). However I didn't measure the angles at the corners (which are in general not 90°), thinking that it would be trivial to find this later.

The first question is: Is it possible to determine the angles from this amount of information? I would say yes as we have 4 knowns (the lengths), and 1 unknown angle (once we have one angle can use cosine rule to get the diagonal and the problem is simple)

If someone can clarify it is possible to find the angles from this much information, that would be fantastic.

Second question: where to go from here? I really want to get this by myself, so no answers (yet). I've playing around with the cosine/sine rules, but with no success. Its not quite as simple as it looks (unless I'm missing something obvious).

Third question: Why don't I just go back and measure the angles? Good point, but its bugging me now...

It'd also be interesting to look at the general case (any number of walls) but I'll try the 4 walls case first!

 

[phyzguy]

You don't have enough information. Imagine taking a small model of your four-walled room. You can bend it back and forth, changing the angles without changing the lengths of the walls. You need to know at least one of the angles, then as you said you can determine the others. Since measuring angles is hard in a big room, try measuring one (or both) of the diagonals. This will then give you the information you need. One trick carpenters use, since they are usually trying to make the angles be 90 degrees, is to measure both diagonals and make sure they are the same. If the diagonals are the same, then the angles are square.

 

[Hemmer]

You don't have enough information. Imagine taking a small model of your four-walled room. You can bend it back and forth, changing the angles without changing the lengths of the walls. You need to know at least one of the angles, then as you said you can determine the others. Since measuring angles is hard in a big room, try measuring one (or both) of the diagonals. This will then give you the information you need. One trick carpenters use, since they are usually trying to make the angles be 90 degrees, is to measure both diagonals and make sure they are the same. If the diagonals are the same, then the angles are square.

I can see why this would be the case for a square or similarly symmetrical shape, as by squashing the square you can mirror the change in angles on each side of the diagonal.

I'm not convinced that this is the case for irregular/asymmetric shapes though? Try it with some different lengths of wire (its important that they are all of different lengths). There only seems to be one possible way to arrange them.

EDIT: I actually measured the diagonals for this purpose - the problem is more just for interest.

 

[phyzguy]

It's still the case with irregular sides. Look at the attached drawing, where I took four different sides and changed the angles.

 

[Hemmer]

Thanks for clearing that up. It does make sense to me now. Thanks for your time!