Compound Miter Calculator

Why Splayed Corners Need Two Angles

A straight-sided box needs one angle at each corner: 45° for four sides, 30° for six. The moment the sides lean outward — a hopper, a flared planter, a tapered lampshade, a feed trough — that single angle splits into two. The cut has to angle across the face of the board (the miter) and through its thickness (the bevel) at the same time, and neither number is simply the corner angle or the slope. Guessing, or cutting 45° on a four-sided hopper, leaves gaps that open wider the steeper the sides lean.

Step by Step: Using the Calculator

Count the sides of the finished piece and decide how far each side leans. Most woodworking charts measure that slope from vertical, so a box with upright sides is 0° and a side leaning out by 15° is 15°; if your plan gives the angle from the floor instead, switch the dropdown to "horizontal". The calculator returns settings for three common setups: the board lying flat on a miter saw table, the board standing on edge with its face against the miter saw fence, and the board lying flat on a table saw with a miter gauge. All sides must be the same width and slope, and every corner is a mitered joint, where both boards get the same cut.

The Formula Explained

With n sides and slope S measured from vertical, the half corner angle in plan view is 180° ÷ n (45° for a square). For a board lying flat on the saw, the blade bevel is asin(cos S × sin(180°/n)) and the miter is atan(sin S × tan(180°/n)). Standing the board on edge against the fence rotates the workpiece a quarter turn, which swaps the roles of the two trig functions: miter = atan(cos S × tan(180°/n)) and bevel = asin(sin S × sin(180°/n)). These match the splayed-box formulas published by Jansson's compound miter calculator and the classic hopper-joint charts in woodworking references. A table saw miter gauge usually reads 90° for a square cut, so its setting is 90° minus the flat-board miter angle. On a miter saw, both scales read 0° for a square cut.

A Real Example: a Four-Sided Planter at 30°

Four sides at 30° from vertical: sin(180°/4) = sin 45° = 0.7071 and tan 45° = 1. Board flat: bevel = asin(cos 30° × 0.7071) = asin(0.6124) = 37.76°, and miter = atan(sin 30° × 1) = atan(0.5) = 26.57° — the familiar "37.8° tilt, 26.6° miter" from printed hopper charts, or 63.43° on a miter gauge. Standing the same board against the fence instead: miter = atan(0.8660) = 40.89° and bevel = asin(0.5 × 0.7071) = 20.70°.

Edge Cases to Watch For

At 0° slope from vertical the result collapses to a plain box corner (bevel 180°/n with the board flat), and at 90° it becomes a flat picture frame (miter 180°/n, no bevel), which is a good way to sanity-check the setup. Many miter saws only bevel to about 45° or 48°, so shallow slopes on three- and four-sided pieces can exceed the saw's range with the board flat; standing the board against the fence usually keeps both numbers smaller. Butt-jointed corners need different angles from mitered ones and aren't covered here. Saw scales are rarely accurate to a tenth of a degree, so cut a test corner in scrap, and keep the same face of every board toward the fence.