Trigonometry for bending

Every offset is a right triangle. The offset height is the side opposite the angle, the conduit between the two bends is the long side (the hypotenuse), and the run along the original line is the side next to the angle (the adjacent). Three ratios link them, and the multiplier chart in your pocket is just one of them written out.

The three ratios

Sine (sin): opposite ÷ hypotenuse, or offset height ÷ distance between bends.
Cosine (cos): adjacent ÷ hypotenuse, or run ÷ distance between bends.
Tangent (tan): opposite ÷ adjacent, or offset height ÷ run.

Turn the sine upside down and you have the multiplier (the cosecant, 1 ÷ sin). Multiply the offset height by it to get the distance between the bends.

Multiplier and shrink per inch

between bends = height × multiplier = height ÷ sin(angle)

shrink = height × tan(angle/2)

The conduit travels the long side of the triangle but only gains the adjacent side along the run. The difference, hypotenuse − adjacent, works out to exactly height × tan(angle/2), so tan(angle/2) is the shrink per inch of offset.

10°: multiplier 5.76, shrink 0.087" per inch (about 1/16")
15°: multiplier 3.86, shrink 0.132" per inch (about 1/8")
22.5°: multiplier 2.61, shrink 0.199" per inch (about 3/16")
30°: multiplier 2.00, shrink 0.268" per inch (about 1/4")
45°: multiplier 1.41, shrink 0.414" per inch (about 7/16")
60°: multiplier 1.15, shrink 0.577" per inch (about 9/16")

Rolling offsets

When the conduit has to move up and over at once, the true offset is the diagonal, by Pythagoras: √(rise² + roll²). Bend that as an ordinary offset, then turn it so the bends lean out of plumb by atan(roll ÷ rise). A 6" rise with an 8" roll is a 10" offset, rolled 53.13° from plumb.

Worked example

A 6" offset at 30°, by the multiplier method

Offset height: 6"
Angle: 30°
Multiplier: 2.00
Between bends: 12"
Shrink (0.268" per inch): 1 5/8"
Run it covers (adjacent): 10 3/8"

Why the calculator goes further

These formulas treat each bend as a sharp corner. A real shoe bends on a radius, which cuts the corner and uses a little less conduit. With a bender selected, the calculator allows for that from the shoe’s centerline radius, so the same offset on a 1/2" EMT hand bender comes out at 11 15/16" between bends and 1 1/2" of shrink. Pick Multiplier method in the bender list when you want the textbook numbers. The centerline radius page explains the difference.