Almighty Calculator

Compound Miter Angle Calculator

For n sides leaning a degrees from vertical, set the blade bevel to arcsin(cos a × sin(180° ÷ n)) and the miter to arctan(sin a × tan(180° ÷ n)).

°
Miter angle (table or fence setting)
14.51°
Bevel angle (blade tilt)43.08 °
Angle between adjacent walls in plan view90 °

Angles are for a board laid flat on the saw table with its long edge against the fence, and the cut made on both ends with the boards flipped so the pairs mirror. Test on scrap: small errors add up around a joint.

Show the math

Half-angle = 180° ÷ 4 = 45°
Bevel = arcsin(cos 15° × sin 45°) = 43.08°
Miter = arctan(sin 15° × tan 45°) = 14.51°

Rounded the same way as the result above.

How it works

When the sides of a box lean outward, the corner where two sides meet is no longer a simple vertical line. A cut that fits has to be angled two ways at once: the blade tilts (the bevel) and the board turns against the fence (the miter). Both angles come from the number of sides and the lean of the walls.

The formulas come from the geometry of two leaning planes and the plane that bisects them. With no lean the miter is zero and the bevel is half the corner angle. As the walls lean more, the miter grows and the bevel shrinks. A four-sided box with 45° lean, the classic pyramid or hopper, works out to a 35.26° miter and a 30° bevel.

Bevel B = arcsin(cos a × sin(180° ÷ n)) Miter M = arctan(sin a × tan(180° ÷ n)) n = number of sides, a = wall lean from vertical Check: a = 0 gives M = 0 and B = 180° ÷ n

Worked example

A four-sided planter whose walls lean 15° from vertical:

  1. Half-angle = 180° ÷ 4 = 45°
  2. Bevel = arcsin(cos 15° × sin 45°) = 43.08°
  3. Miter = arctan(sin 15° × tan 45°) = 14.51°
InputValue
Number of sides4
Wall lean from vertical (0 = straight up, 90 = flat)15 °
ResultValue
Miter angle (table or fence setting)14.51 °
Bevel angle (blade tilt)43.08 °
Angle between adjacent walls in plan view90 °

Assumptions and limits

  • All sides equal, all leaning the same amount, forming a regular polygon.
  • The board lies flat on the saw table with its long edge against the fence, and the finished edges are horizontal (a hopper or bowl). Other setups swap or repeat the angles.
  • Sign conventions vary by saw: bevel left or right, miter left or right. Always cut a test corner from scrap.
  • Angles are geometric. Kerf, blade runout, and board thickness affect the fit.

Common questions

What miter and bevel do I use for a four-sided box with sloped sides?

It depends on the lean. For sides leaning 15° from vertical, the miter is 14.51° and the bevel is 43.08°. With straight sides the miter is 0° and the bevel is 45°.

What is a compound miter?

A cut made with the saw table turned and the blade tilted at the same time, needed wherever two sloped surfaces meet, such as splayed boxes, hoppers, and crown molding.

What is the angle for a 45° pyramid?

For four sides leaning 45° from vertical, the miter is 35.26° and the bevel is 30.0°.

Why does the math use the lean from vertical?

Because it makes the straight-sided case a lean of 0 and the flat case a lean of 90. If you measured the wall angle from horizontal, subtract it from 90 first.

Sources

  • Derived from the geometry of two adjacent leaning wall planes and their mirror (bisector) plane, and checked numerically against a 3D vector construction (see the tests).
  • Standard woodworking references for polygon boxes list the straight-wall case as bevel = 180° ÷ sides.

Updated 2026-09-30