Every conduit bending card in a tool bag lists the same shorthand for a 30-degree offset: multiply your rise by two to get the distance between marks, and add back a quarter inch for every inch of rise. Punch a 6-inch rise into that and you read 12 inches of spacing and 1 1/2 inches of shrink. The spacing is exactly right. The shrink is a little wrong, and on a deep bend it is wrong enough to leave the run short. Both numbers come off the same piece of trigonometry, and knowing where they come from tells you when the friendly rounded version is safe and when it is not.
The multiplier is a cosecant wearing work boots
The distance between your two bend marks on an offset is the rise divided by the sine of the bend angle. One over the sine is the cosecant, so the "multiplier" printed on the card is just the cosecant of the angle you picked. That single fact explains the whole column:
- 30 degrees: 1 divided by sin 30 is 1 divided by 0.5, which is exactly 2.
- 45 degrees: 1 divided by sin 45 is 1.414.
- 22.5 degrees: 2.613.
- 60 degrees: 1.155.
- 10 degrees: 5.759.
It also explains why 30 degrees is the angle most electricians reach for first. Its multiplier lands on a clean 2 with no rounding at all, so the mental math is nothing and there is no accumulated error hiding in a truncated decimal. The pocket card rounds the rest to 1.4, 2.6, 1.2, and 6. Sine behaves smoothly across this range, so the spacing side of the card is the trustworthy side. The worst offender is 60 degrees printed as 1.2 instead of 1.155, which overstates your spacing by roughly 4 percent, and even that only matters on a tall rise.
This multiplier method isn't something an individual shop invented. It's the hand-bending approach Jack Benfield set down in his conduit bending manual decades ago, the reference a lot of apprentices still learn the trade from, and the one Ugly's condensed into the laminated card that lives in a thousand tool pouches. The card is a summary of the math, not a replacement for it, and summaries round.
Shrink is the tangent of half the bend
Here is the part almost no field reference spells out. The shrink per inch of rise is (1 − cos θ) / sin θ. Run the half-angle identities on that expression and it collapses to something much cleaner: the numerator 1 − cos θ is 2 sin²(θ/2), the denominator sin θ is 2 sin(θ/2) cos(θ/2), and the twos and one sin(θ/2) cancel, leaving tan(θ/2).
So the shrink constant for any offset angle is simply the tangent of half that angle:
- 30 degrees: tan 15 is 0.2679.
- 45 degrees: tan 22.5 is 0.4142.
- 60 degrees: tan 30 is 0.5774.
- 22.5 degrees: tan 11.25 is 0.1989.
- 10 degrees: tan 5 is 0.0875.
That's worth carrying in your head, because it tells you at a glance why shrink punishes steep bends so hard. Spacing grows as the cosecant, which flattens out as the angle climbs. Shrink grows as the half-angle tangent, which keeps accelerating. Double the bend angle from 30 to 60 and your spacing multiplier drops from 2 to 1.155, but your shrink per inch more than doubles, from 0.268 to 0.577. A hard bend costs you far more length than a shallow one, and it costs it in a way the linear-feeling "multiply by a constant" habit hides.
The stamped constants are rounded, and always short
The shrink figures printed on most cards are rounded to friendly tape fractions: 1/16 inch at 10 degrees, 3/16 at 22.5, 1/4 at 30, 3/8 at 45, and 1/2 at 60, per inch of rise. Set those next to the exact half-angle tangents and a pattern jumps out.
| Bend angle | Card shrink (per inch) | Exact tan(θ/2) | Extra shrink the card misses, per 10" of rise |
|---|---|---|---|
| 10° | 1/16 (0.063) | 0.0875 | about 1/4" |
| 22.5° | 3/16 (0.188) | 0.1989 | about 1/8" |
| 30° | 1/4 (0.250) | 0.2679 | about 3/16" |
| 45° | 3/8 (0.375) | 0.4142 | about 3/8" |
| 60° | 1/2 (0.500) | 0.5774 | about 3/4" |
The rounded value is lower than the true value at every single angle. That's not a coincidence of which fractions someone picked. The friendly numbers were chosen to be easy to read on a tape, and easy-to-read landed below the real figure across the board. The practical consequence is one-directional: a run laid out off the card shrinks a hair more than the card told you it would, so it comes up short, never long. You never get free length back from the rounding. You only ever lose a little.
When the rounding actually costs you a knockout
Most of the time none of this matters, and pretending it always does is how you turn a fast trade into a slow one. On a shallow 3-inch kick at 10 degrees, the card is off by well under a sixteenth, which is below what you can even mark on a tape. Round away and move on.
The rounding starts to bite in two situations. The first is a deep offset. Take a single 20-inch rise at 30 degrees, the kind of climb you make going up and over a beam. The card says 0.25 times 20, which is 5 inches of shrink. The exact figure is 0.268 times 20, which is 5.36 inches. That is more than 3/8 of an inch of missing length on one bend, enough to walk a coupling past a strut or slide a box knockout out of reach.
The second is a run that stacks several bends. Three 30-degree offsets going around obstructions in one stick each carry that same fraction of missing shrink, and the misses add in the same direction because the rounding always runs short. Three-eighths here and three-sixteenths there stops being rounding noise and becomes a real position error by the time you reach the far end. The decision rule that falls out of the table is simple: trust the card for shallow angles and small rises, and run the exact number for deep offsets, steep bends, and anything you have to chain.
Why two benders of the same size disagree
There is one more number in a good offset that the trigonometry never touches, and it's the one that quietly ruins otherwise perfect layouts. The multiplier and the shrink tell you where your bends sit relative to each other. The take-up, also called the deduct, tells you where to start the whole thing relative to the end of the pipe, and it is a property of the bender shoe, not of geometry.
A half-inch EMT bender doesn't put the arc of the bend exactly at the mark. It springs the bend forward by a fixed amount baked into the shoe's radius, and that amount differs by manufacturer. Klein, Greenlee, and Milwaukee half-inch heads do not all publish the same deduct, and the generic 5 inches that Ugly's prints for half-inch EMT might be 4 3/4 or 5 1/4 on the tool actually in your hand. Borrow a deduct from a card, or from the last bender you owned, and you recreate the exact "eyeballed it and got it wrong" failure the math was supposed to prevent. The trig can be flawless and the run still lands off, because the starting reference was pulled from a different shoe.
That's the honest boundary of what a formula can do for you. The cosecant and the half-angle tangent are universal and exact. The deduct is a measured property of one specific tool, and the only correct value is the one you confirm on your own bender, once, and then keep using.
Once you have seen that the multiplier is a cosecant and the shrink is a half-angle tangent, the pocket card stops being a set of magic numbers and becomes a set of rounded ones you can choose to trust or override. For a shallow kick, round freely. For a deep offset or a stacked run where a half inch changes where a box lands, run the closed-form figure and set your own bender's deduct rather than a borrowed one. If you would rather read the exact marks and shrink for any of the five trade angles without working the tangent in your head, Bendmark computes the full-precision version and rounds only at the last step, to the nearest 1/16.