I visited a friend who owns an octagonal table, and I thought that the tablecloth on it was a little sub-par, so decided to do something about it. Unfortunately, I would not have access to the table while making the cloth and had only two measurements which I had surreptitiously taken. The length of a single side of the table was 45 cm, and the width of the entire table was 105cm.
Due to the wonders of Pythagoras, I was able to make a tablecloth and get it posted off as a Christmas gift without needing further access to the table!
N.B. the cloth is displayed on a larger, circular table above (see comments on access to the model table) and is not ironed because I forgot to take a picture of the cloth after I had ironed it. I shall update this 'Ible once the recipient has photographed the cloth in situ.
Cloth and sewing stuff
Paper, pencil, ruler and scissors
Pocket Calculator or other
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Step 1: Making an Octagon by Folding. and Joke
As we can see in the first image above, an octagon is just eight identical isosceles triangles.
This shape can be made with a single cut  by folding a sheet of paper in the correct way.
First, fold the sheet in half one way (fourth picture).
Second, fold the sheet in half the other way (fifth).
Third, fold a triangle upwards from the centre of the sheet (sixth).
Fourth, turn the paper over and repeat the triangle folding (seventh).
This yields a shape with a 45 degree angle (eighth).
Mark a distance from the point along one side, mark the same distance on the other side and draw a line between the points (ninth).
Cut along that line (tenth photograph).
Once unfolded, you should have the octagon shown in the second picture above.
I say, I say, I say. What do you get if you cross a mountaineer and a mosquito?
(Punchline at the end of this 'Ible)
 I think that _any_ straight-sided shape can be made with folding and a single cut, which is kind of awesome)
Step 2: The Maths Bit! Sizing the Octagon
OK, so we now know that we can make the octagon from a sheet with just one number!
And we have two, so how easy is that!
Unfortunately, the number we have (length of the base of an isosceles triangle) is not the number we need (the length of the two equal sides of an isosceles triangle of single angle 45 degrees whose base is 45 cm).
Fortunately, various folk worked all that out yonks ago, so all we have to do is steal their methods.
In the diagram above, we have distance 'A' and we want distance 'C'.
The equilateral triangle can be split into two right angled triangles, such as α-β-γ.
We know that the line from β-α is half of the 45cm which we have measured from the table, and we know that the angle at γ is 22.5 degrees (half of 45 degrees).
The sine of an angle in a right-angled triangle is equal to the opposite side over the hypotenuse side, so
sin(22.5) = 22.5/(length 'C').
Multiplying each side by (length 'C') makes that
sin(22.5) * (length 'C') = 22.5
Dividing each side by sin(22.5) gives us
(length 'C') = 22.5 / sin(22.5)
Plugging that into a calculator gives 58.8cm.
We are almost there!
We want a bit of overhang over the side of the table, and we want about two cm for making a hem, so we add twelve cm for that.
Therefore the length we want to mark and cut on the fabric will be 58.8 + 2 + 12 = 72.8 centimetres! YAY! Who would have thought that school trigonometry would actually be useful?
Step 3: Folding and Cutting
By now well practiced in the folding arts, I took the fabric, and folded it the same as I had the paper (first few photographs).
Then I marked a point 72.8cm from the point on each side a drew a line between them, using blue chalk for visibility on the mainly white cloth.
Lastly I used a fairly agricultural set of shears to cut the fabric. Since there were eight layers of cloth in total, I actually only cut half of the layers at a time.
If you are using thinner or more delicate fabric, or care more about how the finished product looks, then I would recommend only cutting a couple of thicknesses at a time, possibly remarking the line each cut.
At the end of this process, there is an eight-layer triangle of the correct size.
Step 4: Pinning and Sewing
I went all round the edge of the cloth folding it over twice and then pinning the hem into place.
Once that had been done (which took forever), I ran the cloth through the sewing machine to sew it in place.
I had no idea what type of thread to use, but the local haberdashery was very happy to look at the fabric I had bought, hear the application for which it was intended, and sell me the correct thread.
At this point, it would have made sense to iron the piece and photograph it. I did iron it, but I forgot to photograph it. Apologies.
Step 5: Lessons Learned and Punchline
The first and most important lesson was actually implemented in this 'Ible having been learned the hard way while sewing the cushion for my Pew:- to wit "use pins". I put the zip into the cushion there while just aligning things by hand and it wasn't great. This time, the pins worked. They were a nuisance to put in, and a pain (literally) to take out, but the meant that I could concentrate on sewing while at the machine, rather than dividing my attention with rolling the hem in.
The lesson learned was about orientation in the machine. For some completely unfathomable reason I sewed the material with the good side up, meaning that the hem was on the underside. Gold star if you spotted that in the previous step. There were a couple of places where the hem wasn't quite held perfectly by the pins and where the sewing therefore missed it. It wasn't a biggie, as I found them afterwards and resewed over the affected small lengths, but it could have been avoided.
(reminder of joke, "what do you get when you cross a mountaineer and a mosquito")
It is not defined! You have to use the _dot_ product when you are multiplying a scalar and a vector!
(In the absence of a "Groan" feature on Instructables, feel free to favorite instead ;-)