How to lay out a cone or frustum by hand

A cone or frustum rolls flat into a ring sector: two arcs from one centre, closed by two straight lines. All you need are the two radii and the angle — and one measurement across the sector fixes it exactly, with no stepping round the arc.

Published 2026-10-30. Rather have the numbers worked out? Cone / Frustum Configurator does it in your browser.

What you are making

The worked example is a frustum — a cone with its tip cut off — with a 600 large end, a 300 small end and 400 between them. For plate, work to the mid-thickness diameters; the cone tool can convert inside or outside dimensions for you.

Figure 1. Elevation. The slant side is a true length; extended, the two sides meet at the apex, which becomes the centre of the pattern.

1. Slant height

In the elevation the sloping side shows its true length (Coey, Part III). By Pythagoras, with the difference in radii and the height: √((300 − 150)² + 400²) = 427.20.

2. The two pattern radii

Extend the sides to the apex. By similar triangles the outer radius — apex to large end, along the side — is the large radius times the slant height over the difference in radii: 300 × 427.20 ÷ 150 = 854.40. The inner radius is that less the slant height: 427.20. A full cone has no inner radius.

3. The sector angle

The outer arc has to be as long as the large end's circumference, so the angle is 360° × the large radius ÷ the outer pattern radius — Atkins's rule for a cone, with the frustum's extended slant (Atkins, p. 104): 360 × 300 ÷ 854.40 = 126.40°.

4. Lay it out

From one centre swing both arcs, 854.40 and 427.20, a little longer than you need. Draw one radial line from the centre across both. Now, instead of stepping round the arc, fix the other end with one measurement: the straight distance between the ends of the outer arc is 2 × R × sin(angle ÷ 2) = 2 × 854.40 × sin 63.20° = 1525.28. Measure it across from where the first line cuts the outer arc, and draw the second radial line to the centre. Check across the inner arc: 762.64.

Figure 2. The pattern. One chord measurement fixes the second radial edge; the centre is the apex of the cone.

Why not step round the arc?

The books divide the plan of the large end into spaces and step them onto the outer arc: Coey into twelve (Ch. 20 and 22), Daugherty into sixteen (Problem 43), Kittredge into “as small a space as is convenient” (Problem 122). Take twelve, as Coey does. Each space is a chord of the base circle, 155.29 here, and a chord is shorter than its arc (Kidder, p. 66): twelve of them reach only 125.14° of the 126.40° needed, leaving the pattern about 18.88 short at the seam. Kittredge's advice is to step "as small a space as is convenient". If you do step, use the spacing measured on the pattern — 156.86 for twelve spaces here — or skip stepping and use the chord across the whole sector, as above. Atkins notes that working from the rules usually gives more accurate results than measuring along the arc (p. 105).

5. Check before you cut

The outer arc should measure π × 600 = 1884.96 along the curve, the inner arc π × 300 = 942.48, and each radial edge the slant height, 427.20. Add any seam or lap allowance outside the pattern; none is included here.

Too big for one plate?

Split the pattern into identical segments. Each keeps both radii and takes an equal share of the angle — here three segments of 42.13° — and is laid out the same way: two arcs from one centre, one radial line, and the chord across the segment's outer arc, 614.26, to fix the other. One segment needs a piece of plate about 614.26 × 455.76. The cone-in-segments tool finds the fewest segments that fit your plate.

When the drawing gives a slope, not a height

Drawings often give the side slope instead of the height. With the difference in radii ΔR, a side at an angle α above horizontal rises ΔR × tan α over the cone and runs ΔR ÷ cos α along the slant; the sector angle then comes out as 360° × cos α. So a cone with 60° sides always unrolls into exactly half a ring, whatever its size, and one with 45° sides into 360° × 0.7071 = 254.56°. The cone tool takes the slant height, the side slope or the apex angle directly.

Common mistakes

Using the vertical height as the pattern radius — it is the slant height, measured to the apex.
Reading a cone angle from the wrong line: a slope from horizontal, a wall angle from the axis and an apex angle are three different numbers for one cone.
For a frustum, swinging the two arcs from different centres. Both come from the apex.
Working to the outside diameter of heavy plate: the pattern comes out too long. Use the mid-thickness.
Stepping plan chords round a long arc and finding the seam does not close.

How this matches the tool

The cone tool computes the same radii, angle and chord in closed form, draws the arcs as true arcs in the DXF, and can also do the thickness conversion and press-brake bend lines. Every number on this page was produced by the tool's own code when the page was built.

Sources

Skip the arithmetic

Cone / Frustum Configurator — Flat pattern with inside/outside dimension basis, plate-roll or press-brake forming, bend angles and out-of-round — plus a clean DXF.

Free, in your browser; nothing you type is sent anywhere. How the tools are checked is on the Verification page. Rather have the part drawn for you? Drafting for small shops, quoted per job.