The Geometry of Chairs

A chair leg usually leans in two directions at once. Seen from the side it has front-to-back rake; seen from the front it has side-to-side splay. The drill, however, can enter the seat only once: along one line and at one tilt. Chair geometry converts those two design views into one repeatable boring setup.

The two outputs are:

• The sightline: the direction of the lean when drawn on the seat. You stand or sight along this line while boring.
• The resultant angle: the leg’s true lean away from the seat’s vertical axis.

This lesson uses angles measured from vertical. Some plans and bevel gauges express the complementary angle from the seat surface. If this lesson gives a resultant \(R\), that surface angle is \(90^\circ-R\). Always identify the convention before touching the drill.

Why ordinary addition fails

Imagine one unit of vertical drop from the mortise. The leg moves \(\tan r\) units along the rake axis and \(\tan s\) units along the splay axis. Those movements are perpendicular components, so combine them with the Pythagorean theorem:

The sightline direction, measured in the seat plane from the rake axis toward the splay axis, is:

Using atan2 rather than a simple inverse tangent preserves the correct quadrant when a leg points forward rather than back, or left rather than right. If you are calculating a single magnitude with both components positive, \(\tan\theta=\tan s/\tan r\) gives the same result. Schwarz supplies practical rake, splay, resultant, and sightline guidance in the free Stick Chair Book; Galbert develops the drilling and reaming system in greater depth ^{1 2}.

Worked example: equal rake and splay

Let \(r=10^\circ\) and \(s=10^\circ\).

1. \(\tan 10^\circ \approx 0.1763\).
2. \(\tan R=\sqrt{0.1763^2+0.1763^2}\approx0.2494\).
3. \(R=\arctan(0.2494)\approx14.0^\circ\) from vertical.
4. Equal components place the sightline halfway between the axes: \(\theta=45^\circ\).

The result is 14°, not 20°. Compound leans add like the two legs of a right triangle, not like numbers in a column. That is the central mnemonic: components square; they do not sum.

Turn the angle into a footprint

The same triangle answers two useful shop questions. If the vertical distance from the mortise reference plane to the floor is \(h\):

For \(h=17\) inches and \(R=14^\circ\), the foot lands about \(17\tan14^\circ=4.2\) inches from the mortise in the resultant direction, and the straight-line leg distance is about \(17/\cos14^\circ=17.5\) inches. In our equal-component example, the 4.2-inch plan offset decomposes into roughly 3 inches of rake run and 3 inches of splay run.

These are geometry checks, not a replacement cut list. A real leg needs extra stock for its tenon, leveling, and any foot treatment. Seat thickness, mortise location, and an uneven saddle also affect which reference height belongs in the calculation.

A fast sanity check

Draw a plan-view rectangle for the seat and plot the projected feet. The resulting contact polygon should look intentional and approximately mirrored. If one rear foot appears forward of its mate or one side is dramatically wider, suspect a sign, quadrant, or sightline-measurement error before suspecting the design.

Bench method: Line, Lean, Look

The calculation becomes a drilling routine. Remember Line, Lean, Look:

1. Line: Draw the sightline through the mortise center and extend it far enough to stand over. Mark which end points toward the foot. A direction line without an arrow can be bored 180° backward.
2. Lean: Set a bevel or other reliable guide to the required tilt. Under this lesson’s convention, a resultant 14° from vertical is 76° from a flat seat surface. Follow the plan’s convention if it differs.
3. Look: Align the bit with the sightline in one plane while matching the bevel in the other. Pause after the bit starts and check again. A mirror placed on the sightline can make the second view easier to monitor ².

For tapered joints, a reamer can refine a bored hole and bring the socket toward the target direction. But reaming is controlled removal, not an undo command. Cutting more from one side also shifts or enlarges the opening at the seat. Mark the high side, take a small number of turns, test the leg, and stop when the result is correct. Matching taper geometry between reamer and tenon cutter matters more than the tools’ appearance.

Rule-of-thumb workflow for a published plan

When you are building a proven chair, use the plan’s sightlines and angles rather than recalculating them for sport. When you are changing mortise locations, seat height, rake, or splay, run the numbers and redraw the feet. A practical sequence is:

1. Copy the seat outline and mortise centers full size.
2. Copy each sightline, including its arrow and measurement baseline.
3. Record the angle convention beside every sightline: “14° off vertical” or “76° off seat.”
4. Plot the projected foot locations using the intended height.
5. Make a scrap drilling block at the same thickness and test the setup.
6. Bore the actual seat only after the scrap leg points into the expected quadrant.

Common mistakes and what they teach

• Adding rake and splay. A 10°/10° leg is not 20° off vertical. Use the resultant formula or the plan’s table.
• Measuring the sightline from the wrong edge. “45°” is incomplete unless you know the reference axis and direction.
• Mixing vertical and surface angles. Label the convention; do not rely on what a bevel “looks like.”
• Mirroring the line but not the arrow. Opposite legs usually require reflected directions, not four identical pencil marks.
• Letting a reamer rescue a large error. A reamer can tune a joint; aggressive correction changes the mortise and weakens the fit.
• Cutting legs to seat height. The leg is a hypotenuse and also needs working allowance. Use the plan length, then level the assembled chair.

Geometry is not there to make a simple chair academic. It lets you separate design from luck: draw where each foot should land, convert that choice into one line and one lean, then reproduce it at the bench.