Joinery

Mitring boards of different widths

Why 45° won't close the joint, how to mark the cut without numbers, what to add to the lengths, and how to set the mitre saw.

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A mirror frame: sides from 50 mm stock, top and bottom from 70 mm stock. Every end cut at 45°, the outside corners close, and on the inside of each corner there's a triangular hole, 20 by 20 mm. A 45° cut only suits pieces of the same width.

A 70 mm piece and a 50 mm piece both cut at 45°, the outside corner closed and a triangular hole with 20 mm sides at the inside corner
Both pieces at 45°. The inside corner is missing a 20 by 20 mm triangle.

Mark one piece on the other

The joint line runs from the outside corner of the frame to the inside corner, the point where the inner edges of the two pieces meet. The easiest way to find it is to lay the pieces on top of each other.

Lay the narrow piece across the end of the wide one, as they'll sit in the frame, and check the corner with a square. On the wide piece, mark the outside corner and the spot where the inner edges cross. Lift the narrow piece off and join the two points. Then lay the wide piece on the narrow one and mark that the same way.

Two steps. First the 50 mm piece lies across the end of the 70 mm piece, and a line on the 70 mm piece joins the outside corner to the point where the inner edges cross. Then the pieces swap and the same line is drawn on the 50 mm piece
1: narrow on wide. 2: wide on narrow. Each time, the line runs from the corner to where the inner edges cross.

The finished marks follow a simple rule: the line meets the inner edge as far from the end as the other piece is wide. On the 70 mm piece that's 50 mm, on the 50 mm piece it's 70 mm. Further pieces you can mark with a tape alone.

Two marked pieces: on the 70 mm piece a line from the outside corner to a point 50 mm from the end along the inner edge, on the 50 mm piece to a point 70 mm from the end
The finished marks. Each piece gets the other one's width.

The same rule gives you the lengths. A 400 × 600 mm opening for the mirror needs a top and bottom of 400 + 2 × 50 = 500 mm and sides of 600 + 2 × 70 = 740 mm, measured along the outer edge.

On the mitre saw

With 70 and 50 mm, the wide piece goes at 35.5° and the narrow one at 54.5°. Together they always make 90°. The narrow piece needs more than 45°, and a typical mitre saw only cuts to 45 or 50°.

The maths

wide piecearctan(b / a)

narrow piece90° − arctan(b / a)

a is the width of the wide piece, b of the narrow one. For 70 and 50 mm: 35.5° and 54.5°.

A simple jig helps here: a base with a V-shaped notch that holds the piece turned through 45°. With the saw at zero it cuts at 45°, so for 54.5° you set 9.5°. How to build it is in Cutting past 45° on a mitre saw. Without the jig, cut the narrow piece by hand just beside the line, on the waste side, and plane down to the line. Try it on two offcuts first.

A mitre saw jig from above, saw fence along the top: a base with a V-shaped notch whose faces run at 45° to the fence, a narrow piece lying along one face of the notch, a dashed line where the blade cuts at zero, and the blade turned 9.5° about the middle of the saw fence, just behind the tip of the notch, cutting a small offcut from the end of the piece
The piece lies at 45° to the fence. Turning the saw 9.5° adds the rest, 54.5° in all.

In solid wood

The first problem is the glue. A mitre glues on end grain, which makes a weak joint, so in solid wood it's strengthened with a spline, a thin strip glued into a slot across the joint.

The other problem is wood movement. Solid wood shrinks and swells across the grain, so a board changes width. When the house dries out, the board gets narrower and the mitre opens a gap on the inside of the corner. The wider the board, the bigger the gap. There are three ways out:

  • narrower boards,
  • a butt joint instead of a mitre,
  • plywood or MDF, which barely move.

Sources

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