Tube Bend Length Formula: Arc, Tangent and Cut Length Math

You’ve drawn the part. You know the leg lengths. Now you’re holding a 20-foot stick of tube and a saw, wondering how long to cut it. Add up the legs and the part comes out too long. Guess and you waste tube. The fix is the tube bend length formula: a short bit of math that tells you exactly how much tube each bend uses.

The good news? It’s not hard. You need three numbers: the die’s centerline radius, the bend angle and your leg lengths. From those you get the arc length, the tangent length and the cut length. Once you’ve done it twice, it takes about a minute per part.

This guide walks you through it step by step, the way a friend in the next garage bay would. Every example uses real numbers, and the arithmetic is shown so you can check it. We’ll cover single bends, multi-bend hoops and bumpers, U-bends, marking the tube and the real-world fudge factors that textbooks skip.

The tube bend length formula, explained simply

Here’s the whole idea in one sentence. A bend cuts a corner off your part, so the tube follows a curve instead of a sharp corner. The curve is shorter than the two sides of the corner it replaces. So the cut length is the sum of your legs, minus what the curve saves.

Written as a formula, for legs measured to the corner points:

• Cut length = (sum of all leg lengths) − (2 × tangent length for each bend) + (arc length for each bend)

Stainless Headers’ tube cut length calculator describes the same thing from the other direction: add up the straight sections between bends, then add every bend length. Both ways give the same answer. They just start from different measurements.

The terms you need

Everything is measured along the tube’s centerline, the imaginary line down the middle of the tube. Here are the five terms we’ll use. The example column is a 90° bend on a 6″ centerline radius die.

Term	What it means	Formula	90° on 6″ CLR
CLR	Radius of the bend to the tube’s centerline. Set by the die.	From the die	6.000″
Arc length	Tube used inside the bend. Also called developed length or bend length.	CLR × angle in radians	9.425″
Tangent length	From where the bend starts to the imaginary sharp corner	CLR × tan(angle ÷ 2)	6.000″
Corner point	Where the centerlines of two legs would meet if there were no bend	From your drawing	—
Bend deduction	How much shorter the tube is than the two legs added up	2 × tangent − arc	2.575″

A note on “setback.” Some calculators use it to mean the tangent length. Others use it for a different distance. You’ll also hear “gain” and “take-up,” and trades use those words differently too. To keep things clear, this guide sticks to tangent length and bend deduction. If you’re new to CLR itself, start with our guide to centerline radius.

Step 1: arc length (and why radians)

The arc length is the amount of tube that ends up inside the bend. It’s a slice of a circle’s circumference, measured at the centerline. The formula is simple: arc length = CLR × angle in radians.

Converting degrees to radians

Radians are just another way to measure angles. A full circle is 360°, or 2π radians. So to convert, multiply degrees by π ÷ 180, which is about 0.017453. A 90° bend is 90 × 0.017453 = 1.5708 radians. A 45° bend is 0.7854 radians.

Example: 90° on a 6″ CLR die. 6 × 1.5708 = 9.42″. On a 4.5″ CLR die, it’s 4.5 × 1.5708 = 7.07″. On a 7″ die, it’s 7 × 1.5708 = 11.00″.

The rounded constants you’ll see

Tool makers often round the constant. Stainless Headers uses 0.01745. JD2’s Model 32 manual writes bend length as CLR × degrees × .0175. That’s rounded up about 0.27%, so it reads slightly long. Over one bend, it doesn’t matter. Over 20 bends of 90° on a 6″ die, it adds up to about half an inch.

Constant used	Where you’ll see it	90° on 6″ CLR	Verdict
π ÷ 180 = 0.0174533	Exact value	9.4248″	✅ Use this in spreadsheets
0.01745	Stainless Headers calculator	9.4230″	✅ Fine for shop work
0.0175	JD2 manual	9.4500″	Fine for single bends, reads slightly long
0.001745	A typo seen on some web pages	0.9423″	❌ Off by 10×. Never use

Here’s a handy shortcut. Work out your die’s inches per degree once: CLR × 0.017453. A 6″ CLR die uses 0.1047″ per degree. A 5.5″ die uses 0.0960″. A 6.5″ die uses 0.1134″. Write it on the die. Then any bend’s arc length is just degrees × that number.

Every bend uses tube. The arc length tells you how much, and it only takes one multiplication.

Step 2: tangent length

The tangent length tells you where a bend starts and stops. Picture the two straight legs of a bend. Extend them until they meet in a sharp corner. The tangent length is the distance from that corner back to the point where the tube starts curving.

The formula is tangent length = CLR × tan(angle ÷ 2). “tan” is the tangent button on any scientific calculator or phone calculator. Make sure it’s set to degrees.

The 90° shortcut

For a 90° bend, angle ÷ 2 is 45°, and tan(45°) = 1. So the tangent length equals the CLR. On a 6″ die, a 90° bend starts 6″ before the corner point and ends 6″ after it. That’s worth memorizing.

Tangent and arc for common angles

Here are the numbers for a 6″ CLR die. For other dies, scale them: multiply by your CLR, then divide by 6.

Bend angle	Radians	Tangent length	Arc length	Bend deduction
15°	0.2618	0.790″	1.571″	0.009″
30°	0.5236	1.608″	3.142″	0.074″
45°	0.7854	2.485″	4.712″	0.258″
60°	1.0472	3.464″	6.283″	0.645″
90°	1.5708	6.000″	9.425″	2.575″
120°	2.0944	10.392″	12.566″	8.218″
135°	2.3562	14.485″	14.137″	14.833″

Notice two things. Small bends barely change the length. A 15° bend saves less than a hundredth of an inch. And past 90°, the tangent length grows fast. At 180° it would be infinite, because the legs never meet. We’ll handle U-bends differently below.

Step 3: cut length for a single bend

Now let’s put it together. For one bend, the cut length is leg A + leg B − (2 × tangent) + arc. Both legs are measured along the centerline, from the tube end to the corner point.

Example 1: a 90° bend

You need a 90° bend in 1.75″ tube on a 6″ CLR die. One leg is 24″ and the other is 18″, both measured to the corner point.

1. Tangent length: 6 × tan(45°) = 6 × 1 = 6.000″.
2. Arc length: 6 × 1.5708 = 9.425″.
3. Add the legs: 24 + 18 = 42.000″.
4. Subtract two tangents: 42 − 12 = 30.000″.
5. Add the arc: 30 + 9.425 = 39.425″, or about 39-7/16″.

Check it another way. The straight part of the 24″ leg is 24 − 6 = 18″. The straight part of the 18″ leg is 18 − 6 = 12″. Add the arc: 18 + 9.425 + 12 = 39.425″. Same answer. ✅

Example 2: a 45° bend

Same die, but a 45° bend with two 20″ legs to the corner point.

• Tangent: 6 × tan(22.5°) = 6 × 0.41421 = 2.485″
• Arc: 6 × 0.7854 = 4.712″
• Cut length: 40 − (2 × 2.485) + 4.712 = 40 − 4.971 + 4.712 = 39.742″, or about 39-3/4″

See how the 45° bend only shortens the part by about a quarter inch? The 90° bend shortened it by more than two and a half. Bigger angles save more length.

Multi-bend parts, worked in full

Most real parts have more than one bend. The method doesn’t change. Add every leg, subtract two tangents per bend and add one arc per bend. The only new skill is getting your leg lengths right.

Example 3: a hoop with two 90° bends

Say you’re building a bed bar or a handrail hoop from 1.75″ tube on a 6″ CLR die. The drawing says it’s 42″ wide outside-to-outside and 30.875″ tall from the cut ends to the top of the tube.

First, convert to centerline dimensions. The width is measured on the outside of both legs, so subtract one full OD: 42 − 1.75 = 40.25″. The height goes to the top of the tube, so subtract half an OD: 30.875 − 0.875 = 30″. Your legs are 30″, 40.25″ and 30″.

• Sum of legs: 30 + 40.25 + 30 = 100.25″
• Tangents: 2 bends × 2 tangents × 6″ = 24″
• Arcs: 2 × 9.425″ = 18.850″
• Cut length: 100.25 − 24 + 18.85 = 95.10″, or about 95-1/8″

Here’s where each mark goes, measured from one end of the tube:

Mark	How it’s figured	Distance from end
Start of bend 1	30 − 6	24.000″
End of bend 1	24 + 9.425	33.425″
Start of bend 2	33.425 + (40.25 − 12)	61.675″
End of bend 2	61.675 + 9.425	71.100″
End of tube	71.100 + (30 − 6)	95.100″

Example 4: a bumper with two 30° bends

A simple front bumper has a 40″ center section and two 16″ wings, angled back 30° at each end. All three lengths are to the corner points. Same 6″ die.

• Tangent per bend: 6 × tan(15°) = 6 × 0.26795 = 1.608″
• Arc per bend: 6 × 0.5236 = 3.142″
• Cut length: 72 − (4 × 1.608) + (2 × 3.142) = 72 − 6.431 + 6.283 = 71.852″, or about 71-7/8″

U-bends and bends past 90°

For a 180° U-bend, the legs never meet, so there’s no corner point. Measure each leg to where the bend starts instead. Then cut length = leg A + leg B + arc. Two 20″ legs on a 6″ die: 20 + 20 + (6 × 3.1416) = 58.85″. The legs will sit 2 × CLR = 12″ apart, center to center. For bends between 90° and 180°, either method works, but the tangent lengths get long. Double-check the corner points on your drawing.

Real-world adjustments the formula skips

The formula gives you a perfect, textbook tube. Real tube and real benders add a few wrinkles. None of them are hard to handle once you know they’re there.

Trim allowance

Most fabricators cut a little long and trim to fit. Stainless Headers’ J-bend optimizer suggests roughly 1/4″ of extra length for fit-up trimming. It’s cheap insurance against a short part.

Clamp room

A rotary-draw bender needs straight tube to clamp onto. If a bend sits close to the end of the tube, there may not be enough straight left to grip. The fix is to cut extra length, bend, then trim the excess. Your bender’s manual tells you how much straight its clamp block needs.

Springback and stretch

Tube springs back a little when you release the handle, so you over-bend to hit your angle. The Fabricator notes that harder tube and a tighter CLR mean more springback and more “radial growth.” Bending software can also account for elongation. In plain terms, the real tube may not follow the die perfectly. The fix is a test bend. Bend a scrap piece, measure the real angle and leg lengths, then adjust. Our springback guide walks through it.

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A tool that makes test bends easy: a digital angle gauge

Marking the tube and bending in order

Good math only helps if the marks land in the right place. Here’s a simple routine that works on any rotary-draw bender.

1. Measure the real OD. Use calipers. Convert any outside dimensions on your drawing to centerline dimensions.
2. Do the math on paper. Write down every leg, tangent, arc and the cut length. Add your trim allowance.
3. Cut and deburr. Square cuts make your measurements honest.
4. Draw a reference line. Run a straight line along the whole tube. It shows if the tube rotates in the die.
5. Mark from one end only. Measure every mark from the same end, like the table in Example 3. Measuring mark-to-mark adds up small errors.
6. Line up with the die. Each bender has its own rule for where the start-of-bend mark sits on the die. Follow your bender’s manual.
7. Bend in order. Work from one end. Check each angle with a gauge before moving to the next bend.

Step 6 trips up a lot of people. Rogue Fab’s Bending 101 calls it the bend offset: the distance between the start of the bend and where you line the tube up in the machine. Rogue Fab lists offsets by machine and die, and notes that its tables are being updated. Other makers explain it in their own manuals. Find yours, and confirm it with a test bend.

For parts with bends in more than one plane, you’ll also need to rotate the tube between bends. That’s a topic of its own. See our guide to tube bend rotation, and our step-by-step guide to laying out tube bends.

Real-life lessons from the cut list

These are composite stories. The same situations come up again and again in garage-fab forums and build threads.

Common cut-length mistakes (and how to fix them)

Most wasted tube comes from a handful of mistakes. Here they are, with the fix for each.

Pro tips for faster, more accurate cut lengths

These habits turn cut-length math from a chore into a two-minute job. They also save a lot of tube.

• 💡 Write inches per degree on each die. CLR × 0.017453. Then arc length is just degrees × that number.
• 💡 Memorize the 90° rule. On a 90° bend, the tangent length equals the CLR, and the arc is CLR × 1.5708.
• 💡 Carry four decimals, round once. Do all the math in decimals, then round the final cut length to the nearest 1/16″.
• 💡 Draw centerlines. Sketch each part as a centerline with the corner points marked. Leg lengths come straight off the sketch.
• 💡 Build a cut-list spreadsheet. One row per part, with columns for legs, angles, tangents and arcs. You’ll reuse it on every project.
• 💡 Keep a bend log. Record tube, die, springback and how far your test parts landed from the math. Use it to fine-tune.
• 💡 Nest parts on the stick. Add up your cut lengths plus saw kerf and trim. Plan which parts come from which stick before you cut.

Building exhaust instead? The same math works for mandrel bends. Our exhaust tube cut length guide covers pre-bent U-bends and J-bends.

Frequently asked questions

Your cut-length checklist

Run through this list before you cut any tube for a bent part.

The tube bend length formula is just three small steps: arc, tangent, then cut length. Do them in order, keep everything on the centerline and check with a test bend. Your parts will come out the right size, and your scrap pile will shrink. 🔧