Cone Flat Pattern Calculator
Develop a cone or frustum into a flat sheet layout: outer and inner radii, arc angle, chords and multi-segment patterns, drawn to scale, from diameters and height.
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About this toolWhat is the Cone Flat Pattern Calculator?
Rolled cones turn up everywhere in fabrication: hoppers, reducers, chimney transitions, funnels, cyclone bodies and lamp shades. Each one starts as a flat annular sector, a slice of a ring, cut from sheet or plate and rolled until the straight edges meet. This calculator works out that flat shape from the finished cone’s diameters and height.
Enter the large and small diameters (zero for a pointed cone) and either the vertical height or the slant length. Say whether the diameters are inside, outside or mean, and give the plate thickness; the pattern is laid out on the mean surface so the rolled part comes out at the right size. You get the outer and inner layout radii, the arc angle, the chord across the arc ends, the arc lengths, the half apex angle and the size of sheet the blank needs. Large cones can be split into equal segments, and a to-scale drawing shows the pattern with the segment cut lines.
- Full cones and truncated cones (frustums)
- Height or slant length input, mean, outside or inside diameters
- Layout radii, arc angle, chords, arc lengths and pattern area
- Equal segments with chord and bounding box per piece
- To-scale SVG drawing of the flat pattern
How to use the Cone Flat Pattern Calculator
- Choose mm or inches and enter the large and small diameters. Use 0 for a full cone with a point.
- Choose whether you know the vertical height or the slant length, and enter it.
- Say whether the diameters are mean, outside or inside, and give the plate thickness if they are not mean.
- Enter the number of segments if the blank is too large for one sheet or one roll.
- Read the layout radii, arc angle and chords, check the drawing and copy the numbers to the shop.
Worked examples
Hopper transition, mean diameters
Input
D1 1,000 mm, D2 500 mm, height 500 mm
Output
R 1,118.03 mm, r 559.02 mm, arc angle 161.00°, outer chord 2,205.4 mm
The half apex angle is 26.57°, and 360° x sin 26.57° = 161°. The blank needs a sheet about 2,205 x 1,026 mm.
Same cone in three segments from 6 mm plate, outside dimensions
Input
D1 1,000, D2 500 outside, height 500, t 6 mm, 3 segments
Output
Mean D1 993.29 mm, R 1,110.53 mm, r 551.52 mm, each segment 53.67° with an outer chord of 1,002.6 mm
The diameters drop by 6 / cos 26.57° = 6.71 mm to the mean surface. Each segment fits in about 1,003 x 618 mm.
Pointed cone, the 3-4-5 check
Input
D1 600 mm, D2 0, height 400 mm
Output
Slant 500 mm, R 500 mm, arc angle 216°
With a radius of 300 and a height of 400 the slant is exactly 500, and sin α = 0.6 gives 216°. A pattern over 180° is laid out with the angle, not the chord.
Background and insight
Unrolling a cone is exact, not an approximation, because a cone is a developable surface: it can be flattened without stretching. Every straight line from the apex down the side becomes a radius of the flat pattern, so the slant distance from the apex to the large end becomes the outer layout radius R and the distance to the small end becomes the inner radius r. Their difference is the slant length of the frustum.
The arc angle comes from matching circumferences. The large edge of the pattern must have arc length π x D1, and on a radius R that takes an angle of 360° x D1 / (2R). Since D1 / (2R) is the sine of the half apex angle, the arc angle is simply 360° x sin α. A steep cone with a small half angle has a narrow pattern; a flat cone with a large half angle has a pattern close to a full ring.
Thickness matters for heavier plate. The neutral surface of a rolled plate is near its middle, so the pattern should be developed on the mean diameter. Measured horizontally at the same height, the mean diameter is the outside diameter minus t / cos α, not minus t, because the wall is inclined. For thin sheet the difference is negligible; for 10 mm plate on a shallow cone it is several millimetres of circumference.
Common use cases
Hoppers and bins
Lay out the conical bottom of a silo, feed hopper or dust collector.
Duct reducers and stacks
Make concentric round-to-round reducers and chimney cones from sheet.
Funnels and spinning blanks
Cut blanks for funnels, lampshades and decorative cones.
Plate rolling quotes
Get the blank size and number of segments before ordering plate.
Tips and limitations
- Add seam allowance, weld gap and trim to the net pattern according to your procedure.
- For heavy plate, use mean diameters; the calculator converts outside or inside dimensions for you.
- When the arc angle is over 180°, the outer chord is no longer the widest dimension. Use the bounding box to size the sheet.
- Mark the apex point on scrap or a layout board when the radius is long; a trammel or string swings the arcs.
- Eccentric (offset) cones are not developed by this formula. They need triangulation.
Frequently asked questions
How do you calculate the flat pattern of a cone?
Find the half apex angle α from tan α = (D1 − D2) / (2H). The outer layout radius is D1 / (2 sin α), the inner is D2 / (2 sin α), and the arc angle is 360° x sin α.
How do I lay out a truncated cone on sheet metal?
Mark an apex point, swing arcs of the outer and inner layout radii from it, step off the arc angle (or the chord), and draw the two radial lines from the apex to close the shape.
Should I use the inside or outside diameter for a cone pattern?
Use the mean diameter, at the middle of the plate. Mean diameter = outside − t / cos α = inside + t / cos α, where α is the half apex angle.
What is the slant height of a frustum?
The length along the side from the small edge to the large edge: √(H² + ((D1 − D2) / 2)²). It also equals the outer layout radius minus the inner one.
Can a cone pattern be more than 180 degrees?
Yes. Shallow cones with a half apex angle over 30° need more than half a ring, up to nearly 360° for a very flat cone. Split it into segments to save material.
How many pieces should a large cone be made from?
Enough that each segment fits on a standard sheet and through the rolls, usually two to four. Equal segments make rolling and fit-up easier.
Is there an in-depth guide to the Cone Flat Pattern Calculator?
Yes. “How to Lay Out a Cone or Truncated Cone Flat Pattern” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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