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Right triangles on the job

Two things on a job are right triangles in disguise: a corner you need to prove square, and an offset in a pipe or a cable tray. Both come down to one sentence — the square on the long side equals the two squares on the short sides added together — and one shortcut that lets you skip the squaring entirely.

The one sentence: a² + b² = c²

a — a leg (the run) b — a leg (the rise) c — the hypotenuse (the travel) c = √(a² + b²)
The long side is always opposite the right angle. Square the two short sides, add, take the square root — and you have it.

Three worked examples

1. Squaring a layout with 3-4-5

  1. From the corner, mark 3 ft along one line and 4 ft along the other. (Use 6 and 8 for a bigger, more accurate check.)
  2. Measure the diagonal between the two marks.
  3. If it reads exactly 5 ft (or 10 with the doubled marks), the corner is square. Long, the angle is open; short, it is closed. Diagonal = 5 → square.
Why it works

3² + 4² = 9 + 16 = 25, and 25 = 5². A triangle whose sides fit a² + b² = c² must have a right angle between a and b — that is the theorem run backwards. Any multiple works too: 9-12-15, 30-40-50.

2. The travel of an offset — rise 6", run 8"

  1. The rise (how far over) and the run (how far along) are the two legs. The pipe between the bends is the hypotenuse.
  2. c = √(6² + 8²) = √(36 + 64) = √100.
  3. 10" of pipe between the bend centres. (A 3-4-5 in disguise: 6-8-10.)

3. Where the 45° multiplier comes from — rise 7"

  1. At 45° the two legs are equal: rise 7", run 7".
  2. c = √(7² + 7²) = √(49 + 49) = √98 = 9.9".
  3. Notice 9.9 ÷ 7 = 1.414. That ratio is the same for every 45° offset — it is √2, and it is the multiplier the trade quotes. travel = rise × 1.414.
And the other angles

At any angle, travel = rise ÷ sin(angle). sin 30° is 0.5, so the multiplier is 2. sin 60° is 0.866, so 1.155. sin 22.5° is 0.383, so 2.613. You never need to derive them at the bender — but when you can, you stop mixing them up. The offset math drill practises rise × multiplier.

Try one: legs of 9" and 12" — the long side?

√(81 + 144) = √225 = 15". A 3-4-5 scaled by three.

Drill it

Two legs, find the long side — or the long side and one leg, find the other. Within 1% counts. About half the questions are 3-4-5 families so you learn to spot them.

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This drill needs JavaScript to generate and mark questions. Turn it on to practice — the explanation below works either way.

Questions people ask

What is the 3-4-5 rule?

A triangle with sides of 3, 4 and 5 — in any unit, and any multiple — has a perfect right angle between the 3 and the 4. Measure 3 along one wall, 4 along the other, and if the diagonal between those two points is exactly 5, the corner is square. 6-8-10 and 9-12-15 are the same rule at a size you can actually measure on a floor.

Why does the offset multiplier for 45° come out to 1.414?

Because at 45° the rise and the run of the offset are equal, so the travel is the hypotenuse of a right triangle with two equal legs. Pythagoras: travel² = rise² + rise² = 2 × rise², so travel = rise × √2 = rise × 1.414. Every offset multiplier is that same idea at a different angle — 1 divided by the sine of the angle.

Do I need trigonometry for this?

For 3-4-5 and Pythagoras, no — just squaring and a square root. For the angle multipliers, the sine of an angle is where the number comes from, but on the job you use the multiplier, not the sine. Knowing where it comes from is what stops you misremembering it.

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