Making an Octagon

Hello everyone,
As part of our stadium project we had to make an octagon out of wood, and we thought some of you would want to know how to do it for your projects... So here we go:

First we can determine that the angles within an octagon add to 1080 degrees from the picture below:

There are 6 triangles and since the angles within a triangle add to 180, we get 6x180 = 1080.

Since there are 8 equal corners in an octagon, each interior angle will be 1080/8 = 135 degrees

Now to get 2 pieces of wood to come together at a 135 degree angle, we can see in the picture below that their is a 45 degree gap that must be addressed:

So if we want to cut each board evenly, we divide the 45 degrees in half and set a mitre saw to cut each piece at 22.5 degrees. This will result in the boards coming together at an angle of 135 degrees.

Finally we have to decide on the dimensions of our octagon. Once we decide on the overall length (Ltot) and width (Wtot) we want, we can treat the 4 corners as right angle triangles and assign a length or a width for one of the perpendicular segments (L1 and W1) and solve for the length of the diagonals as per the diagram below:

Equations:

Ltot = L1 + 2*L2;  where L2 = 2*sin(45)*D

Wtot = W1 + 2*W2; where W2 = 2*cos(45)*D

Example:

Lets assume we want our octagon to be Ltot=18 inches by Htot=14 inches and lets also assume we want all 4 diagonal pieces to be the same size as the vertical piece (W1=D). We would get:

Wtot = W1 + 2*W2; becomes 14 = W1 + 2*cos(45)*D; or because W1=D we get:

14=D*(1+2cos(45)), and solving for D gives 5.8 inches

We can now sub that back into our Ltot equation to get the last length, L1

Ltot = 18 = L1 + 2*sin(45)*D; which after subbing 5.8 in for D gives:

18 = L1 +2*sin(45)*5.8

Leaving L1 = 9.8 inches.

And there we have our Octagon. One important thing to note when doing your measurements. Because you are cutting your boards at an angle, the inside of the dimensions of the octagon are going to be smaller than the dimensions of the outside of the octagon. So when doing your measurements be consistent, either measure and cut all of your octagon based on the inner dimensions you desire, or based on the outer dimensions you desire. Do not mix and match!

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