Side a is across from angle A. Side b touches A. Side c is across from the 90° corner.
The ratios behind SOH CAH TOA. The angle card follows angle A from the triangle above until you type your own.
csc, sec, cot are the flips: 1 ÷ sin, cos, tan. Multiply a known side by one of these to get a length.
sin⁻¹, cos⁻¹, tan⁻¹ run a ratio backward to its angle, like the inverse keys on a calculator. The −1 means "undo," not "1 ÷". sin⁻¹ and cos⁻¹ only accept ratios from −1 to 1. Know two sides instead of a ratio? Use the triangle solver above — it picks the right function for you.
On a circle of radius 1, cos is how far across and sin is how far up. Tan is the slope of the line.
All marks are measured from the pipe end. Bends are numbered from the end: Bend 1 is nearest the end, Bend 2 is at the obstruction. In the field you usually mark Bend 2 first (distance to obstruction + shrink), then measure back toward the end by the distance between bends for Bend 1.
Rise and roll are at right angles, so the true offset is their hypotenuse. Bend it as a normal offset of that depth, rotated off plumb by the roll angle.
Two back-to-back offsets at half the center angle. Exact: outer marks = H × csc(θ/2) from center; center mark moves ahead by H × tan(θ/4). Field rule printed on benders (45° saddle): outer marks 2½ × H, center mark 3/16″ per inch of height. Bend order B → A → C: arrow on the outer marks, saddle notch on the center.
Two offsets of depth H with a straight run of W between them. Bend 2 lands at the near edge and bend 3 at the far edge. Add any clearance you want to H and W before you enter them.
Exact values from csc θ and tan(θ/2). Field rules are rounded; at 45° and 60° the exact shrink is a little more than the rule. Geometry only: use your bender's deduct and gain for 90s and stubs.
Arc length is the developed length of conduit in a bend, measured on the centerline radius.
Measure straight across any curve (a sweep, a curved wall, a tank) for the chord, then the gap at the middle for the sagitta. r = (c² ÷ 4 + h²) ÷ 2h.