Formula How To Make A Cone Out Of Sheet Metal
Content Menu
● Core Geometry That Actually Matters on the Floor
● Common Problems and Fixes I See Every Week
● Advanced Stuff the Researchers Are Doing
● Q&A – Stuff People Actually Ask Me
Cones are everywhere in fabricated equipment. Hoppers, cyclones, reducers, stack caps, tank roofs, rocket nose sections—anything that has to change diameter smoothly over a height uses a cone or a frustum. The sheet metal shop gets the job because the part is developable: it can be laid flat, cut from plate, rolled, and the single longitudinal seam welded. Do the layout right and it fits first time. Do it wrong and you spend the night grinding gaps or adding filler strips.
The method has not changed much since the 1940s, but the tools have. Most places now let software do the heavy math and send the file straight to the laser cutter or plasma table. The operator still has to know what the numbers mean, because when the part is 3 m diameter and the roller is only 2.5 m wide you have to split it into gores the smart way.
Core Geometry That Actually Matters on the Floor
Start with the slant height L. That is the only length that never changes from flat pattern to finished cone.
For a full cone (pointed top): L = √(H² + R²) where R is the base radius.
For a frustum (truncated cone): L = √(H² + ((R₁ – R₂)²)) R₁ = large radius, R₂ = small radius.
Everything else in the layout comes from L.
Flat Pattern – The Sector (or Annular Sector)
The flat blank is a portion of a circle whose radius is the slant height L (or the two slant heights for a frustum).
The arc length of the large base must equal the finished large circumference πD₁. The arc length of the small base must equal πD₂.
Because both arcs are struck from the same centre, the sector angle θ in degrees is:
θ = 360° × (R₁ / R_sector)
where R_sector is the radius to the large arc.
For a full cone R_sector = L and θ = 360° × (R / L)
For a frustum the radius to the large arc becomes longer than L:
R_sector large = L × (R₁ / (R₁ – R₂)) R_sector small = L × (R₂ / (R₁ – R₂))
The angle is the same for both arcs:
θ = 360° × (R₁ / R_sector large)
That is the formula you will find in every old sheet metal handbook and in every modern nesting program.
Worked Example 1 – 1200 mm Ø to 300 mm Ø Reducer, 900 mm High
Large diameter 1200 mm → R₁ = 600 mm Small diameter 300 mm → R₂ = 150 mm Height H = 900 mm
Difference in radius = 450 mm L = √(900² + 450²) = √(810000 + 202500) = √1012500 ≈ 1006 mm
Radius to large arc = 1006 × (600 / (600-150)) = 1006 × (600/450) = 1341 mm Radius to small arc = 1006 × (150 / 450) = 335 mm Sector angle θ = 360 × (600 / 1341) ≈ 161.3°
Cut an annular sector: outer radius 1341 mm, inner 335 mm, included angle 161.3°, add whatever seam allowance your shop uses (usually 10–15 mm overlap or butt + strap).
Worked Example 2 – Full Cone 2 m Diameter Base, 1.5 m High
R = 1000 mm, H = 1500 mm L = √(1500² + 1000²) = √(2250000 + 1000000) = √3250000 ≈ 1803 mm θ = 360 × (1000 / 1803) ≈ 199.7°
Almost 200° sector with 1803 mm radius, arc length checks: (199.7/360) × 2π × 1803 ≈ 6280 mm = π × 2000 mm. Perfect.
Worked Example 3 – Offset (Eccentric) Cone
When the small opening is not centred, slant length varies around the circumference. You cannot use one clean sector any more. The practical way is to divide the large and small circles into 12 or 24 equal parts, calculate the true slant length to each point (triangulation), plot the points on the flat, and connect with straight lines (gore method) or smooth curves if you have software. Most CAD packages now have an “unfold” or “sheet metal” command that does it automatically from the 3D model.
Cutting the Blank
Small cones – laser or plasma straight from the nested dxf. Large cones that do not fit the table – split into 6–12 gores, cut individually, add lap on the radial edges. Very thick plate (≥10 mm) – often oxy-fuel or waterjet because laser slows down.
Leave the correct seam allowance from the start. A lot of shops add the allowance on the pattern itself so the cutter knows exactly where the weld prep starts.
Rolling the Cone
Standard Three-Roll Benders
Feed the blank large-arc first for shallow cones, small-arc first for steep cones. The top roll is tilted so its centreline is parallel to the cone generator.
The operator watches the radial chalk lines (drawn every 100–200 mm) and keeps them parallel to the roll axes. If the small end starts to run ahead, he snubs it with a chain or a come-along until the lines stay straight.
Typical minimum small diameter on a given machine is roughly 1.5–2 times the top roll diameter. Below that you get wrinkling on the inside unless you have cone rolling attachments or you pre-form the small end.
Splitting into Gores for Big Diameters
Anything over about 3–3.5 m finished diameter usually has to be made in petals because no plate is wide enough and no roller long enough. Eight to sixteen gores is common. Each gore is rolled as a partial cone, then the longitudinal seams are fitted and welded on the floor using strongbacks to hold curvature.
Press-Brake Segmented Cones
For very thick material (12–25 mm) or very steep angles where rolling would buckle the compression flange, many shops brake-form each gore in 100–200 mm bumps and weld the radials. It is slower but gives perfect geometry.
Material Thickness Change
Pure bending (ideal rolling) keeps thickness almost constant. In practice you lose 3–10 % on the small end because of circumferential compression. If you spin or incremental-form the cone you can lose up to 50 % following the sine law t_final = t₀ sinα where α is the semi-vertex angle. That is why research papers on SPIF and explosive forming always quote the sine law – their processes are stretch-dominated.
Common Problems and Fixes I See Every Week
- Seam opens or overlaps → slant height calculated wrong or material thickness ignored in layout (use mid-plane or inside/outside depending on code).
- Small end wrinkles → too much compression; slow the feed, increase bottom roll crown pressure, or relieve the leading edge before rolling.
- Large end goes barrel-shaped → top roll pressure too high in the middle of the pass.
- Cone twists during rolling (eccentric) → no centreline marked or the blank was not fed square.
Advanced Stuff the Researchers Are Doing
Some jobs cannot be rolled the conventional way – huge single-piece rocket sections or high-strength alloys that crack in three-roll bending. That is where explosive forming and single-point incremental forming come in. Detonate a charge over a blank clamped on a die and you get a perfect cone in one shot. SPIF uses a simple hemispherical tool on a CNC mill or robot and walks spiral paths – no die needed, great for prototypes. Thickness still follows the sine law, so you start thicker than the finished minimum.
Final Check Before Welding
Roll the cone, tack the longitudinal seam every 150–200 mm, then measure circumference at three heights with a π-tape. Tolerance is usually ±0.2–0.5 % of circumference depending on code (ASME, EN, or customer spec). Check height and verify the small end is perpendicular to the axis with a square or laser level.
Once it passes, run the root pass, back-gouge if full penetration is required, and finish.
That is the whole story from drawing to finished cone. The formulas are simple Pythagoras and similar triangles, the skill is in the rolling and in knowing when to split the part or change the method. Do it enough times and you can look at a sketch, do the numbers in your head, and tell the programmer exactly what the flat pattern has to look like before breakfast.
Q&A – Stuff People Actually Ask Me
- Q: The nesting software gave me a sector over 3 m radius but my laser table is only 2 m × 6 m. What now?
A: Split into 4–6 gores automatically (most programs have a “segment cone” button), add 15 mm lap on radials, nest the petals. - Q: Customer wants 1.4301 stainless 4 mm thick, 2400 mm to 600 mm in 1200 mm height. Will it roll clean?
A: Yes, but expect 8–12 % thinning on the small end and plan the seam allowance on the thick side. Roll slow, keep rolls clean. - Q: Why does the cone come out 5 mm taller than drawing?
A: You laid out on the centreline but rolled to the inside surface. Subtract half thickness × (large + small perimeter difference)/perimeter in the height calc next time. - Q: Can I make a cone from one piece if the sector angle is only 90°?
A: Yes, shallow cones often end up 80–120° sectors. Just make sure your roller can take the big radius blank. - Q: Best way to do a 5 m diameter silo roof cone in 6 mm plate?
A: 12–16 gores, roll each on the 3-roll, fit on the ground with spiders and strongbacks, weld radials first, then circumferential rings for stiffness.