Calculating interior angles of an irregular hexagon?
September 20, 2014 11:26 PM   Subscribe

I'm trying to create a 3D model of a six-sided area of my garden. I know the lengths of each side. Is it possible to work out the exact angle of each corner? Here's a link to a rough picture - note that the labels are correct but the sides and angles aren't to scale.

Going clockwise, the lengths are:

A - 4.2 metres
B - 4.3 metres
C - 6.16 metres
D - 1.1 metres
E - 3.5 metres
F - 4.6 metres

I've tried to rough it out in Sketchup and with pieces of dried spaghetti cut to size but I can't be precise. Googling tells me that the sum of the internal angles is 720* in a hexagon, and further Googling reveals a number of triangle angle calculators, but nothing for a hexagon.

(It's been a long, long time since I studied any math, so I hope I've got the terms right.)
posted by UltraFleece to Science & Nature (8 answers total) 2 users marked this as a favorite
If you can also measure the diagonals (i.e. make a triangle using two sides and one diagonal), you could use the law of cosines to find the missing angles. The side lengths alone are not enough to uniquely determine the angles.
posted by ktkt at 11:49 PM on September 20, 2014 [5 favorites]

As the problem is set up, you don't have enough information to calculate the angle of each corner.

Simplifying, it is as if you'd laid four identical length segments to make a rhombus (diamond shape); you don't know if the figure is a square, or has two 30 degree angles and two 150 angles, or what.

The easy way is to make more measurements - measure the distance of the obtuse triangle with the other two sides A and B, and with this information use trig or a formula to get the angle between segments A and B. Rinse, repeat.
posted by sebastienbailard at 11:49 PM on September 20, 2014

Unless you know something else, like a couple of the angles, I think you're out of luck. Hexagons, like quadrilaterals (but unlike triangles) are not rigid.

Think about this as an experiment. Take four sticks and make them into a square by attaching their ends together. You can flex the shape to form a rhombus and you can even squish it all the way flat. So for quadrilaterals and hexagons, side length doesn't force particular angle choices.
posted by leahwrenn at 11:50 PM on September 20, 2014

Oh, you have access to the garden and the issue is just making a model in the computer? Yeah, I agree with ktkt, just measure some diagonals so that you're making pairs of adjacent sides into triangles, whose angles are forced by side lengths.

(Or, go buy a cheap protractor and just measure the angles in the garden. It'll be close enough given the precision of your measurements.)
posted by leahwrenn at 11:54 PM on September 20, 2014

The short answer is, no you can't work out the angles just from knowing the lengths of the sides of a hexagon. The reason is, there are a whole bunch (literally, an infinite number) of different solutions that all have the exact same lengths of sides, but different angles.

Probably the simplest way to solve this is to think in terms of triangles. Unlike objects with more than three sides, a triangle IS completely determined if you know the lengths of the three sides. So you have to somehow divide your entire figure into triangles and then measure each of the sides of each of the triangles.

One method for doing this: Measure three additional interior distances so that the entire figure is subdivided into triangles--something like this.

Then you can use the ruler/compass method outlined below to draw triangle ACE, because you know each of the lengths AC, CE, and EA. With that triangle ACE as the basis, use the same method to add the two remaining sides and complete triangle ABC. Similarly complete triangles CDE and EFA. And there you go!

Another method: Pick a point somewhere near the center of the hexagon and measure the distance from that central point to each corner of the hexagon. Now you've neatly divided the whole hexagon into six triangles and you know the length of each side of all six triangles. Example image (this is a regular hexagon, but you can get the idea).

Once you have everything divided into triangles, by whatever method you choose:

* You can put the lengths of the sides into a triangle calculator like this or this to get the angles.

* Once you know the lengths of each of the three sides, it is easy to draw the triangle using a compass and ruler. This little animation shows how.
posted by flug at 12:10 AM on September 21, 2014 [2 favorites]

Is the garden shape on google maps satellite imagery? If so could you take a screenshot out and use that as a template.
posted by 92_elements at 6:32 AM on September 21, 2014 [1 favorite]

The law of cosines us your friend. You can google it. You need to measure the distance across the garden from corner so that whichever angle you are evaluating is part of a triangle. You will need a calculator (or google I suppose) that computes arccosines.
posted by SemiSalt at 8:28 AM on September 21, 2014

Once you have the length of the diagonals, you can draw intersecting circles with the appropriate radii in sketchup, no trigonometry required.
posted by signal at 7:23 PM on September 21, 2014

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