Lesson 9 of 13
Angles That Are Not 90
Trim carpenters are separated from the rest by two habits: they stop assuming the corner is square, and they know the difference between the angle of the cut and the number on the saw. Those are not the same number, except at ninety degrees, which is why so many people never find out.
Step one
Read the material
Three ideas. The middle one is the trap, and it has cost a great deal of trim.
Concept one
A closed shape divides 180 by its sides
Build a picture frame, an octagonal column wrap, a hexagonal planter. For any regular closed shape with equal sides, the number you set on the miter saw is 180 divided by the number of sides.
Four sides: 180 over 4 is 45. Eight sides: 180 over 8 is 22.5, which is why that number is printed on every saw ever sold. Six sides: 30. Five sides: 36. Twelve: 15.
Carry that one division and you never need a table. Every closed shape you will ever cut is in it.
Concept two
The cut face is half the corner. The saw is not.
Two pieces meeting in a corner share the angle between them, so the cut face on each one meets the board's edge at half that corner. An 88 degree corner gives a cut face of 44 degrees to the edge. That part of the folk wisdom is correct.
The trouble is the next step. A miter saw does not measure the cut face against the board edge. It reads zero at a square crosscut and counts how far you have swung away from square. So the number it wants is 90 minus half the corner.
At a true 90 degree corner, half the corner is 45 and 90 minus 45 is also 45. The two numbers land on top of each other, the shortcut appears to work, and nobody learns the difference. Walk into an 84 degree corner using the shortcut and you will cut 42 when the saw needed 48, and the joint will open by six degrees.
Concept three
Tighter corner, bigger number
There is a sanity check that costs nothing and catches the error every time. Ask which way the number should move.
A tighter corner has more material to remove, so the saw swings further from square and the setting goes up. A wider, more open corner needs less removed, so the setting goes down. An octagon's corner is a wide 135 degrees and sets at a small 22.5. A sharp 60 degree corner sets at 60.
If you compute a setting and it moved the wrong direction from 45, you have used half the corner where you needed 90 minus half the corner. Cut a test piece in scrap, hold the halves together in the actual corner, and trust the joint over the arithmetic and over the label on the saw.
A note on saws. Scales differ between makes, and a few are labelled so the number you set is the cut face rather than the deflection. This lesson teaches the common convention, where a square crosscut reads zero and a picture frame reads 45. Cut scrap on any saw you have not used before and let the joint settle the argument.
Step two
Work it with your hands on it
Set the corner you measured. The panel gives you both numbers side by side, so you can watch them separate as soon as you leave ninety.
Hands on it, off the screen. Take an angle finder to four corners in your own house and write down what you actually get. Most people are surprised the first time.
Then cut two scraps at the saw setting this panel gives you, hold them together in that corner, and look at the joint against the light. That is the whole lesson in about ten minutes.
Step three
Prove you understood it
One question per idea, new numbers every time.
All three solid.
You can miter a closed shape, tell the cut face from the saw setting, and check that the number moved the right direction. Lesson ten is circles and arcs.
Trades Geometry
Lesson nine of thirteen. Written by our founder, Dr. Gene A Constant, and donated to the Foundation. Published by Global Sovereign University Press.
The book page Buy a copy — it funds a free seatGSU is a 501(c)(3) nonprofit and book sales are its only income. A purchase is not a tax-deductible donation. As an Amazon Associate, GSU earns from qualifying purchases. This lesson is free and always will be.