How the Chair Leg Angle Calculator works
Turn the rake and splay you want into the two numbers you drill with: the sightline to draw on the seat, and the resultant to set your bevel gauge to, measured off square.
How this is calculated
A leg that leans two ways at once can be drilled with a single tilt, as long as the tilt goes in the right direction. The sightline is that direction, drawn on the seat. The resultant is how far to tilt.
Treat the leg as a line through the centre of its mortise, in a seat that is flat and level, and follow it down one unit of height. Seen from the front, the foot has moved out to the side by tan(splay). Seen from the side, it has moved toward the front or the back by tan(rake). Looking straight down, those two moves are the sides of a rectangle on the seat, and the leg's shadow is its diagonal. That shadow is the sightline: sight along it and the leg has no sideways lean left, so it looks plumb. The angle of the diagonal and its length give the two numbers.
sightline = atan( tan(rake) ÷ tan(splay) ) measured from the baseline
resultant = atan( √( tan²(rake) + tan²(splay) ) ) measured off plumb
and backwards
rake = atan( tan(resultant) × sin(sightline) )
splay = atan( tan(resultant) × cos(sightline) )The convention, pinned. The baseline is the line through the left and right leg of a pair, parallel to the front of the seat. The sightline angle is measured at the mortise from that baseline, opening toward the middle of the seat. A leg with splay and no rake has a sightline of 0°, lying on the baseline. A leg with rake and no splay has a sightline of 90°, running front to back. Equal rake and splay give 45°. The foot ends up on the far side of the mortise from the sightline, away from the middle of the seat: out to the side by the splay, and by the rake toward the front for a front leg or toward the back for a back leg. That is why the tool asks which leg you are laying out: the two numbers are the same, but the drawing is mirrored front to back. When you drill from the underside, the drill stands where the leg will, so it leans the way the arrow in the drawing points.
The resultant is the leg's lean off plumb, so on a bevel gauge it is set off square: the blade stands that many degrees from upright, not that many degrees from the stock.
Some plans dimension the sightline from the other direction. The underside layout in Christopher Schwarz's sawbench article marks the same line as 64° from the line joining the two legs at one end of the bench, the end he calls the front, and 26° from the line along the bench. For those leg angles the formula gives 63.8° and 26.2°. If your plan measures from the centreline or from a long edge, its figure is 90° minus the one shown here.
At the drill. Schwarz's article describes the routine. Mark the leg centres and draw the sightlines on the underside of the seat with a bevel gauge. Reset the gauge to the resultant and fix it on the sightline; he clamps it there. Stand with your feet square to the sightline, put the tip of the bit on the mortise centre and tip the brace until it matches the blade of the gauge. A spotter can tell you whether the bit is staying parallel to the blade. He bores until the lead screw of the auger pokes through, then turns the seat over and finishes the hole from the other face.
How the sources line up. The Black Dog's Woodshop, Woodworking in a Tiny Shop and Matthew Kudija each write the derivation out and reach equivalent formulas, though not in the same printed form; Kudija's page credits the Tiny Shop solution with helping to catch an error in an earlier version. Peter Galbert's note that equal rake and splay always give 45° is the same rule. This calculator reproduces the Black Dog's worked example and every one of the 450 values in the Tiny Shop table for 1° to 15°.
Two figures in the sources were not calculated, and they differ from the formula. Schwarz finds his resultant by leaning a bevel gauge against a bent-wire model and reads about 15° for 14° rake and 7° splay, where the formula gives 15.5°; his 64° sightline for the same sawbench does agree and is used as a test. The Blue Spruce Woodshop post steps its resultants off with a protractor used as a ruler and gets 13° for 10° rake and 7° splay, where the formula gives 12.1°. A comment under that post explains why the method overstates the angle and gives the calculated figures, and the author accepts the correction; the comment's figures are used as tests here and the post's own are not.
What this does not cover. It assumes a flat seat that sits level, takes rake and splay from 0 to 60°, and handles one leg at a time. It does not lay out leg positions, account for a seat that is tilted or saddled where the mortise goes, or give angles for stretchers.
What this calculator is tested against
This tool cannot be published unless it reproduces every example below. 9 are copied from a published source. 9 were worked by hand from the cited rule.
| Example | Expected | Basis |
|---|---|---|
| Rake 12°, splay 14° | sightline 40.4°, resultant 18.1° | Published The Black Dog's Woodshop, worked example |
| Front leg with rake 2°, splay 4° | sightline 26.5°, resultant 4.5° | Published Woodworking in a Tiny Shop, how to read the table |
| Rake 3.4°, splay 6.7° (decimal input) | sightline 26.8°, resultant 7.5° | Published Woodworking in a Tiny Shop, spreadsheet entry box |
| Rake 5°, splay 10° | sightline 26.4°, resultant 11.1° | Published Woodworking in a Tiny Shop, rake and splay table |
| Rake 15°, splay 1° (almost all rake) | sightline 86.3°, resultant 15.0° | Published Woodworking in a Tiny Shop, rake and splay table |
| Rake 14°, splay 7° (the sawbench angles below, as tabulated) | sightline 63.8°, resultant 15.5° | Published Woodworking in a Tiny Shop, rake and splay table |
| Sawbench with 14° rake and 7° splay, sightline found with a wire model | sightline 64°, as marked on the article's layout drawing (its resultant, read off a wire model with a bevel gauge, is "about 15°"; the formula gives 15.5°) | Published Christopher Schwarz, "Compound Angles; No Math", Popular Woodworking |
| Stool front legs: rake 7°, splay 7° | resultant 9.85° | Published Comment under the Blue Spruce Woodshop post, giving the resultants by calculation |
| Stool back legs: rake 10°, splay 7° | resultant 12.13° | Published Comment under the Blue Spruce Woodshop post, giving the resultants by calculation |
| Equal rake and splay (10° each, chosen here as the instance) | sightline 45° | By hand Peter Galbert, Chair Notes: the rule that equal rake and splay always give a 45° sightline |
| Splay only: rake 0°, splay 14° | sightline 0° (along the baseline), resultant 14° | By hand Leg-vector geometry (see How this is calculated) |
| Rake only: rake 10°, splay 0° | sightline 90° (square to the baseline), resultant 10° | By hand Leg-vector geometry (see How this is calculated) |
| Plumb leg: rake 0°, splay 0° | resultant 0°, no sightline | By hand Leg-vector geometry (see How this is calculated) |
| Backwards: sightline 40.4°, resultant 18.1° | rake 12.0°, splay 14.0° | By hand The Black Dog's Woodshop example read in reverse |
| Backwards: sightline 90°, resultant 15° | rake 15°, splay 0° | By hand Leg-vector geometry (see How this is calculated) |
| Backwards: sightline 45°, resultant 20° | rake 14.4°, splay 14.4° | By hand tan(rake) = tan(splay) = tan 20° × sin 45° = 0.3640 × 0.7071 = 0.2574 |
| Round trip: rake 12°, splay 14° converted and converted back | rake 12°, splay 14° again | By hand Leg-vector geometry (see How this is calculated) |
| Round trip: rake 7.5°, splay 22.5° converted and converted back | rake 7.5°, splay 22.5° again | By hand Leg-vector geometry (see How this is calculated) |
Sources
- The Black Dog's Woodshop: Deriving the Formulas for Sightline and Resultant AnglesDerivation from the two right triangles a plumb line makes with the leg, with a worked example: 12° rake and 14° splay give a sightline of 40.4° and a resultant of 18.1°.
- Matthew Kudija: Chair Angle CalculatorDerives equivalent formulas and credits the Woodworking in a Tiny Shop solution with helping to catch an error in an earlier version. Cites Christopher Schwarz's The Stick Chair Book and Drew Langsner's The Chairmaker's Workshop, and reproduces Langsner's diagram of the sighting line measured from the line connecting a pair of legs.
- Woodworking in a Tiny Shop: Resultant and Sightline Angle CalculatorA derivation with a table of sightline and resultant for every whole-degree rake and splay from 1° to 15°. All 450 values are checked by this tool’s tests.
- Christopher Schwarz, "Compound Angles; No Math", Popular WoodworkingDefines rake, splay, sightline and resultant; shows a sawbench with 14° rake and 7° splay whose underside layout marks the sightline 64° from the line across one end of the bench and 26° from the line along it; describes drilling with a bevel gauge set on the sightline.
- Peter Galbert, Chair Notes: Sightlines RevisitedStates that equal rake and splay always give a 45° sightline.
- Blue Spruce Woodshop: How I Understand Sight LinesA stool with 7° splay, 7° rake at the front and 10° at the back. The post steps its resultants off with a protractor used as a ruler (about 10° and 13°). A comment under the post explains why that overstates the angle and gives 9.85° and 12.13° by calculation, and the author accepts the correction. The comment’s two figures are used as tests here; the post’s own are not.