How to Pie Cut Exhaust Tube: Angles, Segment Lengths and Fit-Up

A pie cut bend is built from short, wedge-shaped slices of straight tube, welded back together so the tube turns a few degrees at each joint. It is also called a segmented, mitred (mitered) or lobster-back bend. This guide covers the geometry, the maths a pie cut calculator does, and how to mark, cut, tack and weld a bend that closes up square.

What a pie cut is, and when to use one

Put the slices back together with every short side on the inside and the tube turns at each joint.

Pie cuts beat a mandrel bend when:

• You need a radius you can't buy. Mandrel bends come in the radii your supplier stocks. A pie cut can be almost any radius.
• The tube is big. Large-diameter mandrel bends cost more and are harder to find in tight radii.
• Space is tight. Intakes, turbo inlets and outlets and downpipes often have to turn hard next to the turbo, a chassis rail or the bulkhead.
• You want the look. Rows of even TIG joints are a style of their own.

The cost is time: six joints to cut, fit and weld per 90°. If a stock bend fits, use it. For help choosing a radius, see 1D vs 1.5D vs 2D bend radius.

Pie cut angles: how many joints make a bend

The cut angle is how far one face leans from square. The joint angle is how far the tube turns where two faces meet. Both faces lean, so joint angle = 2 × cut angle.

Our pie sleeves mark 7.5° cuts, so each joint turns 15° and joints = bend angle ÷ 15.

Cut the bend out of one length of tube, as in the method below, and the first and last cuts leave an angled end on the straight tube either side. Those two ends are part of the bend, so the number of loose segments is one fewer than the number of joints: five for a 90°.

Weld the segments between two square-cut ends of tube instead and you need one more: six for a 90°. The joint at each end then turns 7.5°.

Bend angle	Joints (15° each)	Loose segments, cut from one length	Segments between square-cut ends
15°	1	0	1
30°	2	1	2
45°	3	2	3
60°	4	3	4
90°	6	5	6
180°	12	11	12

The same holds at any cut angle. With 11.25° cuts (22.5° joints), a 90° is four joints: three loose segments cut from one length, or four between square-cut ends.

Segment length: the maths behind a pie cut calculator

Want the numbers without the sums? Our pie cut calculator works out the slices, cut angle and side lengths for your tube, bend and radius. The maths it uses is below.

Each segment has a centreline length, a short side (the throat, inside the bend) and a long side (the heel, outside). With R the centreline radius (CLR), D the actual outside diameter (2.5" tube = 63.5 mm; 2" NPS pipe = 60.3 mm) and θ the joint angle in radians (15° = 0.2618):

Centreline = R × θ
Short side = (R − D/2) × θ
Long side = (R + D/2) × θ

Worked example: 2.5" (63.5 mm) tube at 1.5D, so R = 95.25 mm (3.75"), with 15° joints:

• Centreline: 95.25 × 0.2618 = 24.9 mm (0.98")
• Short side: 63.5 × 0.2618 = 16.6 mm (0.65")
• Long side: 127.0 × 0.2618 = 33.2 mm (1.31")

A 90° is six centreline lengths: 6 × 24.94 = 149.6 mm, the arc length of a 90° at that radius. Its centre-to-end is R, near enough, the same as a mandrel bend of that radius.

Tighter radii leave less throat to fit and weld. At 1D on this tube the short side is 8.3 mm; at 0.5D it is zero.

Segment sizes at 1.5D with 15° joints (mm, rounded):

Tube OD	R (1.5D)	Centreline	Short side	Long side
1.5" (38.1)	57.2	15.0	10.0	19.9
1.75" (44.45)	66.7	17.5	11.6	23.3
2" (50.8)	76.2	19.9	13.3	26.6
2.25" (57.15)	85.7	22.4	15.0	29.9
2.5" (63.5)	95.3	24.9	16.6	33.2
2.75" (69.85)	104.8	27.4	18.3	36.6
3" (76.2)	114.3	29.9	19.9	39.9
3.5" (88.9)	133.4	34.9	23.3	46.5
4" (101.6)	152.4	39.9	26.6	53.2
4.5" (114.3)	171.5	44.9	29.9	59.8

How to mark and cut pie cuts

Marking

1. Cut one blank for the whole bend: enough for the bend, both legs, one kerf per cut and something to hold.
2. Draw two alignment lines 180° apart along it, labelled A and B. They become the throat and heel. To find the opposite side, wrap a paper strip round the tube, mark the overlap and fold it in half.
3. Mark the cut positions on the side of the tube, halfway between A and B. A cut crosses that line where it crosses the centreline, so the gaps there are true centreline lengths. Put the first mark half a centreline length past where the bend begins, then step one centreline length plus one kerf at a time.
4. Scribe each cut line round the tube at 7.5°, alternating the lean from cut to cut. The slices come off alternately short side on A and short side on B, with no waste. Check: the gaps along A run short, long, short; along B, the opposite. Without a sleeve, use a printed wrap-round template. An angled cut line unrolls into a sine wave, so a straight strip won't draw it.
5. Number every piece and mark its throat.

Marking with a pie sleeve (pie cut jig)

A sleeve does steps 3 and 4 for you, and gives every line the same lean. Our pie sleeves are plastic rings with both edges leaning 7.5°, so scribing along both edges marks a complete 20 mm slice with 15° between its faces.

1. Draw two lines along the tube 180° apart, on the sides of the bend (halfway between the throat and heel).
2. Slide the sleeve on with its two slots on those lines. It's a loose fit, so hold it in place.
3. Scribe along both edges. That's one segment.
4. Turn the sleeve 180° round the tube, butt its edge up to the line you just drew and scribe the next slice. Repeat for every segment.

Because every slice comes from the same sleeve, the segments are identical and the bend has no wander. The 20 mm is the narrowest point of each slice, on the inside of the bend. That fixes the short side of every segment, so the sleeve also sets the radius: the centreline radius is 76.4 mm plus half the tube OD (20 mm ÷ 0.2618 = 76.4 mm).

Sleeve (OD tube)	Bend radius (CLR)	As a multiple of OD
1.5"	95 mm	2.5D
1.75"	99 mm	2.2D
2"	102 mm	2D
2.5"	108 mm	1.7D
3"	114 mm	1.5D
3.5"	121 mm	1.4D
4"	127 mm	1.3D
4.5"	134 mm	1.2D

The SCH10S sleeves work the same way, from 93 mm on 1" NPS to 127 mm on 3½" NPS. Each cut takes a blade's width off the shared line, so the finished bend comes out a little tighter: about 4 mm of radius per 1 mm of kerf. For a different radius, use the segment maths above and mark by hand.

Cutting

• Bandsaw. Set a horizontal bandsaw's vice to 7.5°. Cut, turn the tube exactly 180° on your alignment lines, feed it to the next mark and cut again. Turning the tube flips the lean, so the vice never moves.
• Cut-off wheel. A 1 mm stainless disc in an angle grinder, following the line round the tube. Expect more dressing.

Kerf doesn't change the angle, but it shortens every segment and tightens the bend. Add one kerf per spacing and cut down the middle of the line. Wear eye protection.

Deburr and dress the faces

Deburr inside and out; a burr stops the faces closing. Dress each face flat on a linisher without changing its angle. Offer each pair up on its lines and look for light through the joint. Thin exhaust tube wants a tight, square-edged butt, not a bevel.

Fit-up, tacking and welding

1. Turn every second segment 180°. Every short side is now on the throat, and the alignment lines meet at each joint.
2. Build on a full-size drawing of the centreline and both legs, scribed on the bench.
3. Tack the throat and heel first, then the sides. Keep tacks small and the lines matched.
4. Check the angle as you go. Half a degree out per joint is 3° out over a 90°.
5. Tack the whole bend, check it, then weld.

On stainless, TIG with an argon back purge is the usual approach. Plug or tape the ends, purge out the air and keep it flowing while you weld, or the inside of each joint oxidises ("sugaring"). Weld short sections on opposite sides in turn to limit distortion. Use fume extraction.

Rolling segments to change plane

A pie cut doesn't have to stay flat. Roll a segment in its joint (turn it round the tube, faces together) and everything after that joint swings into a new plane by roughly the roll angle. That is how one bend turns and climbs at once, say a turbo outlet clearing a chassis rail.

• The faces still meet. A 7.5° face is an ellipse less than 1% longer than it is wide, so the mismatch is a fraction of a millimetre.
• The rolled joint loses a little angle. Rolled 15°, a 15° joint becomes 14.9°; rolled 30°, 14.5°.

Mark index lines before you tack, so you roll by a measured amount.

Our sleeves have notches every 15° round the rim for exactly this. They let you move the cut round from one side to the other in 15° steps, so the next slice turns the bend into a new plane by a known amount.

Common mistakes

• Not turning alternate segments. Reassembled as cut, the slices make a straight tube.
• Losing the order. Number every piece at the saw.
• Ignoring kerf. Right angle, wrong radius.
• Measuring centreline lengths on the throat or heel. They are only true on the side.
• Grinding the angle away while deburring.
• Checking only at the end. Errors add up over six or twelve joints.
• Heavy tacks on one side. The weld shrinks and pulls the joint open opposite.
• No back purge on stainless. Sugaring can't be fixed from outside.

Tools that help

• Pie cutting sleeves for OD tube, 1.5" to 4.5": both edges at 7.5°, so each fitting marks a slice 20 mm at its narrowest with 15° between its faces, which sets the bend radius for that size. Two reference slots 180° apart, adjustment notches every 15° and a hanging hole.
• Pie cutting sleeves for SCH10S pipe, 1" to 3.5" NPS: the same geometry, for schedule 10 pipe.

Both are in the pie cutting guides collection, made to order in our UK workshop, with other sizes on request. Match the sleeve to your material: tube is sized by actual OD and pipe by nominal bore, so 1.5" NPS pipe is 48.3 mm OD, not 38.1 mm. Unsure which you have? See SCH10S pipe vs OD tube sizes.

FAQ

How many pie cuts do you need for a 90° bend?

With 7.5° cuts, a 90° bend has six 15° joints. Cut from one length of tube, that is five loose segments, because the tube either side keeps an angled end. Welded between two square-cut ends of tube, it is six.

What angle do you cut pie cuts at?

Half the joint angle you want. 7.5° cuts give 15° joints, which divide evenly into 30°, 45°, 60°, 90° and 180°.

How do you work out pie cut segment length?

Centreline = R × θ, short side = (R − D/2) × θ, long side = (R + D/2) × θ, with θ in radians. See the table above for common tube sizes, or let the pie cut calculator do it.

Can I make a bend that isn't a multiple of 15°?

Yes. Make one joint smaller with a hand-marked cut: 50° is three 15° joints plus a 5° joint, cut at 2.5° per face.

Do stainless pie cuts need a back purge?

Yes, or the bore side of every joint oxidises. Mild steel doesn't.