# Hexagon Legs

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## Introduction: Hexagon Legs

A few weeks ago, I wanted to spend some time in my workshop making small stools with some of the wood I had available. For a moment, I wish I had a lathe so I could turn scrap 2x4 into something more visually pleasing. If you have been playing games in the past, you probably realized how frequently the hexagon shape is being used which got me interested in the geometry of this shape. What I realized is that this shape is ideal for leftover 2x4. Let's look into the geometry of the hexagon first.

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## Step 1: Geometry of the Hexagon

So the interesting thing about the haxagon is the relationship of the large radius (R),the small radius (r) and the side length (t). It is not everyday that you read the wikipedia entry for specific shape but it turned out very insightful for the next phase of my work.

Here is my main question : What are the dimensions of the hexagon that fits in the height of a 2x3 or 2x4 ?

For those of you following at home, what are the R and t parameters if r =0.75"

Well basically, R and t are equals in an hexagon which I initially thought was an interesting fun fact. If you remember your trigonometry courses, you can then see that r=R x cos(30).

So I made a few test runs with some 2x3 and kind of fine tuned my table saw but it was a bit onerous. I then ran out of scrap 2x3 and turned to my 2x4. My Eureka! moment was when I checked if two hexagon legs would fit. In other words, 4R = 3.5". OMG this is exactly the width of a 2x4. I am going to have EXACTLY two legs per stock and this means I will be able to optimize my cuts quite nicely. So I RUSHED (I wish I was kidding) to a 2x4 and scribbled the outline of two hexagons on the end of the stock and then went to calculate the setting I would need to make my cuts on the table saw.

## Step 2: First Cuts : Removing the 4 Corners (ingot Phase)

For the first cut, I know that I can fit EXACTLY two hexagons side by side (4R=3.5") but I need to remove the extra. In other words :

4R = 2t + 4C

C = 7/16"

Where C is the leght of the cut I need to take out of each corner.

Since all the angles of an hexagon are 120deg, you want to angle your table saw at a 30 degree angle cut on the right side of the blade. I initially ran with my calculation (3.5 - 7/16) but that did not take into account the width of the blade. I finally removed half of the blade which gave my a first cut at 3-1/32".

At first I did one side and then to the next but as I progressed, I found out the final result was better if I alternated the corners (Bottom left, Top right, Top left, bottom right).

When you are done with this step, you should have your 2x4 with all 4 corners striped at a 30deg angle. It kind of looked like an ingot so I called this step the ingot phase.

## Step 3: Second Cut : Splitting It in Two (teardrop Phase)

That second cut is the most critical part of the project. Now that you have removed the corners, you want to cut the part at exactly the right place. Since they fit quite nicely in a 2x4, you will make 2 cuts in one here. Probably time to take out your best blade with that high tooth count (I used 60 for this project).

You need to take out again that same 7/16" out of the newly splitted parts. Basically you want to end up with t+C or 7/8" + 7/16". Once again, I removed half of the width of the blade and set my table saw to 1-9/32".

When you are satisfied with your cut, you should be left with two parts that kind of look like a teardrop.

## Step 4: Final Cut : Making the Hexagon

This is the best part, you only need to take that final extra length and your table saw is already set at the right mark. Flip the teardrop part over and cut that last section to make your final hexagon.

Voilà ! With only 7 cuts and two marks, you have 2 hexagon shaped legs and also something extra...

After I was satisfied with my settings, I started making a bunch of them for my future projects. When I was doing that last cut, I realized that the offcut from the last step is a perfect equilateral triangle. I have no idea what I am going to do with them but I do know that it would be a pain to make them so perfect even if I wanted to !

You will see as well that some of your legs might be a bit off compared with the others. On the last picture, I show the run of 16 I made. You can see that some aren't as nice as the others. I think this is mostly because my table saw isn't that precise but also that the width of the scraps I used probably differed a bit. Nontheless, that is good enough to use for a later project.

Happy making !

Participated in the
Furniture Contest 2018

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## 11 Discussions

Nice but.... how exactly is this "furniture" to entre the competition (no I don't have an entry).

Talk about chancing your arm. - almost as bad as the kid who screwed 2 brackets to a skatebord and called it a "shelf" without actually fixing it to the wall and entered that.

I hope you can see that this wasn't by any stretch of the imagination a piece of furniture in itself but more something I wanted to share with the community so they can include it in their own project. As for why I entered it, when you complete an instructable, they automatically ask you if you want to include it in of their current contest. Needless to say, I am not holding my breath of any of the top prizes.

From the Furniture Contest page:

"The Furniture Contest is open to all projects furniture-related."

So maybe it's a little stretch, but I would say a tutorial on how to make a part of a piece of furniture counts as "furniture-related."

My calculations seem to be a little different than yours? On your graph paper notes. Where and what is the difference?
D = outside circle = 1.500"
d = inside circle = 1.2990" ~ 1.300"
R = outside circle radius = 0.750"
r = 0.694"
t = 0.75"

You are correct regarding my calculations which are for a single piece of wood for the Octagon calculation. My primary interest is to make Knife Handles for Japanese cutlery, so all I need is one handle for each knife. I will be using some stock from exotic wood, which cost a little more than a Pine 2 x 4. For these projects. I'll be using a single square piece of wood. Which will produce an equilateral Octagon.

By-the-way your notes helped me quite a bit.
Nice project you presented.

Thomas

Trying to do an octogon was my logical next step. If you do end up making one, I would really like to see it !

I have generated an Excel program, still need to find any bugs. If you are interested in a copy, let me know. Any feedback would be appreciated.
Thomas

Your question got me to check my numbers again. I'll add that some of my notes that would appear on the pictures were refined later when I figured that I could fit 2 legs in on 2x4.
I have added an extra picture so that it is a little bit clearer what I ended up with.

In your numbers, it looks like you don't have the hexagon oriented so that the small diameter (d) in the height of the 2x4 (1.5")

Seems to me that the spare triangles would be useful for stands for painting or finishing. Also within a few seconds and a bandsaw you could turn them into the actual pyramid shapes that are so popular for paint stands. Good article when even the end scraps are useful! Al

Thanks for sharing. This will surely come in handy for somebody.

Neat series of set-ups for the saw. Thank you for sharing the design process :-)