Mitre angle calculator: skirting, frames, polygons and square corners

To join two pieces in a corner, each one is cut at half the corner angle, along the line that bisects it: 45° in a square corner, 67.5° in a 135° corner, and half of (n − 2) × 180 ÷ n for a frame with n sides.

A mitre follows the line that halves the corner

Two lengths of skirting, coving or picture-frame moulding meeting in a corner are each cut at half the corner angle. The cut line bisects the angle, splitting it into two equal parts, so the pieces close up without a gap. In a square corner the cut is 90 ÷ 2 = 45°. In a 135° corner it is 67.5°.

On a mitre saw whose scale reads 0° for a square cut, you set the difference from square, which is 90° minus the cut: 45° for a square corner and 22.5° for a 135° corner. If your saw is marked another way, use the angle on the piece, which the calculator also shows.

Skirting: measure the real corner angle

Room corners are rarely exactly 90°. A 92° corner needs cuts of 46°, a saw setting of 44°: cut at 45° and you leave a 1° gap on each side, easy to see on white skirting. Measure the angle with a sliding bevel or a digital angle finder and enter the reading.

The calculation is the same for an internal corner and for an external one, such as a chimney breast or the end of a stud wall: only the side of the piece that stays longer changes.

Mitre angles for frames and regular polygons

Each piece of a closed frame is cut at half the interior angle of the polygon. The last column is the mitre setting measured from a square cut, as shown on a saw whose scale reads 0° for a square cut. Example: an octagon has interior angles of 135°, each piece is cut at 67.5° to its edge, and the saw is set to 22.5°.

Frames and polygons

For a closed frame with n equal sides, the interior angle is (n − 2) × 180 ÷ n, the formula for a regular polygon. Each piece is cut at half that angle, and the saw setting is 180 ÷ n. A hexagon has 120° corners: each piece is cut at 60°, a saw setting of 30°. An octagon needs 67.5°, or 22.5° on the saw, and a pentagon 54°, or 36°.

The outside length of each piece is longer than the inside length by 2 × width × tan(180° ÷ n). For a square frame with 40 cm inside sides in 3 cm moulding, each piece measures 40 + 2 × 3 = 46 cm on the outside. As a hexagon, the same moulding gives 40 + 6 × tan 30° = 43.46 cm per piece, 260.8 cm in all.

Square corners with 3-4-5: the check diagonal

To set out a right angle on site, measure 60 cm along one side and 80 cm along the other: the diagonal between the two marks should be 100 cm. This is the 3-4-5 triangle from Pythagoras, since 60² + 80² = 100². With 3 m and 4 m sides the diagonal should be 5 m, and the bigger the triangle, the more accurate the check.

If the diagonal measures 101 cm instead of 100, the actual angle is 91.2°, worked out with the cosine rule. At 99 cm it is 88.8°. The calculator gives the actual angle and the diagonal difference, so you know which way to adjust.

Roof pitch and rafter cuts

A slope in per cent becomes degrees with the arctangent: 45% is 24.23° and 100% is 45°. The other way round, a 35° roof pitch is a 70.02% slope. The rise and the horizontal run give the angle directly: a 1.50 m rise over a 3 m run is 26.57°.

On a rafter pitched at 35°, the plumb cut, which is vertical, is 35° off square: the piece makes 55° with the edge of the rafter. The seat cut, which is horizontal, is 55° off square. The two settings always add up to 90°.

Mistakes that spoil a mitre

• Cutting at the full corner angle instead of half: in a 90° corner, a 90° cut is just a square end.
• Setting 67.5° on the saw for a 135° corner, when the setting from square is 22.5°.
• Measuring the angle above the skirting, where the plaster rounds off, instead of at floor level.
• Reading a 30% slope as 30°: 30° is 57.74%.

For the next steps, the slope calculator works out the gradient of a ramp or a waste pipe, the wall area calculator gives the length of skirting and the paint area, and the pool volume calculator uses the same site measurements.