Those Legs Ain’t Square

Splay Legged Stool

Starting now on a new build thread, a Japanese splay leg stool. Why a stool you ask? A splay leg saw horse is in the series of exam challenges for a Japanese carpenter after a ‘hopper’, introducing the fundamental concepts of working with compound angles that are necessary for the complexity of understanding and laying out hip roof framing. I merely chose to make a stool as opposed to a saw horse because horses are best made in sets of two, and a stool takes far less lumber. Plus, I’ve felt the need for a short stool to sit upon while sawing lumber (oddly enough two identical stools would be better for this task as well, but one suffices).


I’ve cut a roof in the past where I ran into some of these compound angles and used several modern western books to understand the layout, which has gotten me by just fine. But the Japanese methods for some reason just make more sense. And I want to cut timber frames inspired by Japanese structural joinery.


To avoid some awkward geometry where the rails meet the legs the footprint of the legs are made square to plan (the view seen from above). As a consequence of the leg being splayed, tilted at an angle in two axis, is that the cross section of the leg becomes diamond shaped.

So, how does one figure out the right shape to make the leg? Lets pull back a minute for a bit more introduction to rise/run and the hopper shape.

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Here from the Japanese text “How to use the Sashigane” is a picture of two different hoppers, a mitered hopper and a butt edged hopper. The geometry of this form applies to a splay leg stool as well, which you might come to see as a hopper turned upside down.

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On the left page is the introduction to slope expressed as rise/run, just as you might use in describing a roof in the west as 6/12, or 12/12. For a base unit run of 12 the roof rises, or displace, a given amount, and the pitch of the roof that keeps the rain off your head is defined. In the rest of the world run is defined with a base unit of 10. So in the case of the hopper in the photo above for a base unit of 10 it has a displacement of 3, or a 3/10 slope seen in elevation (side view).

Given that basic relationship a triangle is formed, with the two right angle sides defining the hypotenuse (seen on the right page above). In the Japanese system each of the sides of this triangle is given a name, ko , ko , and gen 玄.

By further dividing the unit triangle the slope angles for compound miter cuts can be found. The first division, cho-gen, is made by drawing a line perpendicular to the hypotenuse (gen) which intersects with the right angle corner of the rise/run. In the text above they help you out by assigning cho-gen a square shape. Knowing this length allows you to form a new triangle that gives the face cut angle on a hopper, or the angle that a rail meets a leg for the stool that I’m making.

By using the base unit of 10 on one arm of the framing square and the cho-gen length on the other, a new triangle is formed and you descend down the rabbit hole of ki-ku-jutsu! Fun times! The sashigane is your calculator, no math required. Similarly the edge miter angle for a hopper (or the edge miter of a rail meeting a leg) is found in the same unit triangle with the division of the hypotenuse by the cho-gen line.

TAJCD Volume 2 cover

Want to know more about this without searching for obscure Japanese carpentry texts? Chris Hall has published a great series of texts here. I’ve only bought the second one, and its helped me tremendously. Help a brother carpenter out and send him some love (and by love I mean money).

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This page shows one good method (far from the only way) to find the correct shape for the leg of my stool. What they’ve done is folded the sides of the leg out like opening a cardboard box and layed it flat to represent it accurately in two dimensional form.

Using the width of the leg along the ground a square box is drawn. Next a line is drawn diagonally that represents the outside arris (corner) to the inside arris. By taking the width of the leg perpendicular to its length with a compass and swinging it down to where it intersects the diagonal the first of the corners can be defined. See? This shit all boils down to basic geometric concepts and you should be able to fill in the rest of the leg cross section from there. The photo above also shows the method for laying out the rail mortises so that they do not interfere, as well as an interesting way to find the angle of the top and bottom edges of the mortise (sho-chu-ko, a further division of the unit triangle).

But how does one draw the foldout view of the leg at the right angle in the first place? It is in fact not the same as a simple elevation view, remember that in elevation the leg is leaning away from you, so the true lengths of the joinery along the leg cannot be taken directly.

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Here is the elevation view, and they’ve done something really cool. Can you tell? They’ve folded the whole drawing on one side up along the ground axis so that the leg face is represented truly flat/vertical! If you were sitting on the ground in front of the stool its like leaning the whole stool towards you. In elevation view the leg is defined by the common slope (3.5/10). By projecting a vertical line along the slope of the leg a point is chosen along the slope and swung with a compass up to meet the vertical line. Because any given point remains consistent along the vertical axis you can now draw the face of the leg so that measurements along its length are accurate.

Notice that the leg is now at a new slope. If you were to take your sashigane and measure this new slope, what would you find? Its our old friend Chu-ko slope! Okay, too much exclamation, but this unit triangle shit is pretty cool. This is the same slope used to draw the foldout view of the leg. There’s more important detail to the above drawing, but I’ll save that for when I get to the rail tenons.


Back in the real world I became a little suspicious of the accuracy of the straight edge on my sashigane, must have dropped it again…So I made a simple wooden straight edge long enough to draw the center lines on my stool legs, about 24″ of vertical grain doug-fir, the same lumber I resawed for the stool.


The two halves are cut from one piece, book matched, and match drilled for dowels that are glued in one half. The dowel is a good tight fit and allows the edges to be planed together.


By planing them when they’re together any deviation from straight is doubled when viewed with their edges together. Its a simple technique and beautifully accurate. Its accuracy that you can really trust because you always have a reference edge to check it against, no wondering how straight it is, check that shit and be sure. The side with the glued dowels is the reference, in practice only the other half is used to check flat and draw straight lines.


I used the same method I explained above to find the cross-section directly on the end grain of the leg, and inked a line on the faces to show how much material to remove. In order to know how close to meeting the other edge I was while planing I used a pencil to darken the wood. When the pencil graphite is gone you’ve planed out to the corner.


Five legs (one spare for when I fuck up), ready for layout, a post for tomorrow. If you think this was complicated enough prepare to join me further down the rabbit hole.

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