Cone layout: flat pattern for a cone or truncated cone (formulas + worked example)

Short answer — the flat pattern for a truncated cone is an annular sector, and it takes four lines of math. Extend the cone to its apex: Hf = H × D_large / (D_large − D_small). Outer pattern radius: R2 = √(Hf² + (D_large/2)²). Inner pattern radius: R1 = R2 × (D_small / D_large). Included angle: θ = 360° × (D_large/2) / R2. For the classic hopper — 24" big end, 6" small end, 18" tall — that gives R2 = 26.83", R1 = 6.71", and θ = 161.0°. Swing those two arcs from one center point, close the sector at 161°, add a seam allowance, and you have the blank. A full (non-truncated) cone is the same idea with R1 = 0: a plain pie-slice sector of radius L = √(H² + R²) and angle 360° × (R / L).

The formula

FULL CONE (base radius R, height H):
  Slant height:      L = √(H² + R²)
  Flat pattern       = sector of a circle, radius L
  Included angle:    θ = 360° × (R / L)

TRUNCATED CONE / FRUSTUM (D_large, D_small, vertical height H):
  1. Extend to apex:   Hf = H × D_large / (D_large − D_small)
  2. Outer radius:     R2 = √(Hf² + (D_large/2)²)
  3. Inner radius:     R1 = R2 × (D_small / D_large)
  4. Included angle:   θ  = 360° × (D_large/2) / R2

  Flat pattern = annular sector (ring segment) between R1 and R2.

SELF-CHECKS (do these before cutting):
  Outer arc length = π × D_large   (it must wrap the big end)
  Inner arc length = π × D_small   (it must wrap the small end)
  R2 − R1          = √(H² + ΔR²)   (the frustum slant, where
                                    ΔR = (D_large − D_small)/2)

Worked example: 24 in to 6 in hopper, 18 in tall

Given:
  D_large = 24"    D_small = 6"    H = 18" (vertical height)

Slant of the frustum (for the R2 − R1 check):
  ΔR    = (24 − 6)/2 = 9"
  slant = √(18² + 9²) = √405 = 20.12"

1. Extend to apex:
  Hf = 18 × 24 / (24 − 6) = 18 × 24 / 18 = 24"

2. Outer pattern radius:
  R2 = √(24² + 12²) = √(576 + 144) = √720 = 26.83"

3. Inner pattern radius:
  R1 = 26.83 × (6 / 24) = 6.71"

4. Included angle:
  θ = 360° × 12 / 26.83 = 161.0°

Checks:
  R2 − R1   = 26.83 − 6.71 = 20.12"  ✓ matches the slant
  Outer arc = 2π × 26.83 × (161.0/360) = 75.4" = π × 24  ✓
  Inner arc = 2π × 6.71  × (161.0/360) = 18.9" = π × 6   ✓

Layout chord (to set the second radial line):
  chord = 2 × R2 × sin(θ/2) = 2 × 26.83 × sin(80.5°) = 52.9"

If it's 1/4" plate and 24"/6" are INSIDE diameters, use mean:
  Mean D_large = 24.25"    Mean D_small = 6.25"
  → Hf = 24.25"  R2 = 27.11"  R1 = 6.99"  θ = 161.0°

Truncated cone (the hopper case)

Almost every real job is a frustum, not a full cone — hoppers, reducers, chute transitions, stack caps, funnel liners. The trick that makes the math fall out is extending the cone to its imaginary apex. A frustum is just a full cone with the pointy end cut off, so its flat pattern is a full-cone sector with a smaller concentric sector removed. That's why you compute Hf first: it locates the apex, the apex-to-big-end slant becomes R2, and the small end sits at R1 along the same rays. The included angle comes from one requirement only — the outer arc of the pattern must equal the circumference of the big end (π × D_large), because that arc becomes the big end when you roll it.

If you're fighting CAD instead of a soapstone — the recurring Onshape-forum question is "how do I flatten a sheet metal frustum for a hopper?" — the same numbers apply: model the cone as sheet metal with a seam gap so the flatten feature has somewhere to open, or skip the wrestling match and construct the annular sector directly as a sketch using R1, R2, and θ. Either way, check the flattened outer arc against π × D_large before you nest it. CAD flatteners apply a K-factor for you; hand math is where it gets forgotten.

Marking it out by hand

You don't need software for a one-off. You need a trammel (or a beam compass — even a stick with two holes drilled at R1 and R2 spacing), a straightedge, and a tape.

• Pick a center point off the corner of the plate — the apex. On the hopper, the whole pattern lives inside roughly 54" × 34", so plan the nest before you scribe.
• Scribe a straight baseline from the apex, longer than R2. This is your first seam edge.
• Swing the outer arc at R2 = 26.83" and the inner arc at R1 = 6.71" from the apex. Use a trammel bar or wire — string stretches and will lie to you by an eighth over a 27" swing.
• Set the second seam edge at θ = 161.0°. Easiest without a protractor: mark the chord — 52.9" from where the baseline crosses the outer arc, swung to intersect the outer arc — then scribe apex-to-that-mark.
• Double-check by walking the outer arc with dividers or a flexible tape: it must read 75.4" (π × 24). If it doesn't, your angle is off — fix it now, not after plasma.
• Add the seam allowance along one radial edge: butt-weld seams need a kerf-plus-gap allowance; lap or lock seams need ½"–1" extra depending on the joint. The development above is net — edge to edge, zero gap.

Rolling the cone

This is where the forum troubleshooting threads live: the pattern was right, but the cone came out lobed, spiraled, or flat-ended. A cone will not roll like a cylinder — the big-end arc has to travel farther through the rolls than the small-end arc, so if you feed it straight, it walks sideways and spirals.

• Pre-bend the leading and trailing edges before rolling. Slip rolls can't curve the last couple inches at each end of the blank; skip this and you get flat spots at the seam that no amount of re-rolling fixes.
• Feed it in an arc. Keep the small end toward the tighter (adjusted) side of the rolls — or use the cone-rolling attachment if your machine has one — and steer the blank so the small-end arc feeds slower than the big-end arc. Short passes, walking the blank around its own apex.
• Tilt the pinch on machines that allow it: set the rolls slightly closer on the small-end side so the tight end takes more curvature per pass.
• Mind grain direction. Roll with the bend axis across the grain where you can; rolling with the grain on hard temper material invites cracking at the tight end, and springback differs with vs. across grain, so a blank nested sideways rolls differently than the test piece.
• Sneak up on it. Springback means the cone comes off the rolls slightly open. Many light passes beat one heavy pass — over-rolled is much harder to recover than under-rolled.

When the plate is too small: gores

A big cone often won't fit on stock plate — the hopper pattern above already needs a 54" swing, and a 60" diameter transition is hopeless as one piece. The fix is gores: divide the development into two, three, or four identical sectors (each with angle θ/n and the same R1 and R2), roll each segment, and seam them vertically. Every extra seam is welding and grinding time, so use the fewest gores that fit your plate and your rolls.

Skip the math

FabCalc does this whole page in one screen: enter the two diameters, the height, and the plate thickness, and it generates the flat pattern with the neutral-axis correction and K-factor applied — then exports a 1:1 DXF for the plasma or laser table with kerf compensation, or prints a tiled paper template at full scale for hand layout. Full cones, frustums, and gored developments, with the arc-length checks built in. Free on iOS and Android, works offline in the shop.

Related

• Heat input for the seam weld — SA-516 Gr 70 root pass limits
• All fabrication calculators
• FabCalc — cone layout + flat patterns on iOS and Android

Note: This is a geometric development — it assumes zero material stretch and ignores springback, so treat it as the starting blank, not gospel. Before cutting plate, verify with a test template: pros print the pattern 1:1 on paper or cardboard, tape it up into the cone to confirm fit, then dot-punch through the paper onto the plate. Reviewed 2026-07.