Cone development

What it is

The truncated cone — the frustum, the concentric reducer — is the transition between two diameters on the same axis. Developed, it becomes an annular sector: two concentric arcs closed by two straight sides. The formula gives the slant height, both radii and the sector angle; it is marked out with a trammel from the centre.

Where it is used

Pipe reducers, funnels, hoppers, stack transitions, silo and cyclone outlets.

Measurements the calculation needs

• D1 Smaller diameter
• D2 Larger diameter
• H Vertical height between the openings
• THK Plate thickness

Calculate with your own measurements

No account needed: enter the measurements in millimetres and get the figures and the drawing. The scaled PDF template, the DXF for the cutting table and the 3D model take a free account.

Smaller diameter D1 (mm)

Larger diameter D2 (mm)

Vertical height between the openings H (mm)

Plate thickness THK (mm)

Calculate free

How it develops: the formulas

Slant height

g = √( h² + ((D2 − D1) / 2)² ) h is the vertical height between the openings, not the slant. For D1 = 800, D2 = 1200 and h = 800: g = 824.6 mm.

Thickness offset, normal to the surface

δ = (t / 2) × h / g = (t / 2) × cos α α is the half-angle, tan α = (D2 − D1) / (2h). The mean-surface radii are r1 = D1/2 − δ and r2 = D2/2 − δ. In the example, in 3 mm plate: δ = 1.46 mm.

Inner sector radius

Ri = g × r1 / (r2 − r1) The distance from the trammel centre to the small opening. In the example: 1643.2 mm.

Outer sector radius

Re = Ri + g = g × r2 / (r2 − r1) In the example: 2467.9 mm. Both arcs are concentric: one trammel centre.

Sector angle

θ = 360° × (r2 − r1) / g = 360° × r1 / Ri In the example: 87.3°. The shallower the cone, the larger the angle; a flat disc gives 360°.

Arc lengths

small arc = π × (D1 − 2δ) ; large arc = π × (D2 − 2δ) For checking the layout: each arc must measure the circumference of its opening on the mean surface.

Worked example

The values the form comes pre-filled with, run through the tool itself:

Smaller diameter	800 mm
Larger diameter	1200 mm
Vertical height between the openings	800 mm
Plate thickness	3 mm
The flat pattern fits a plate of	3407 × 1279 mm

Result of the example

Plate	3407 × 1279 mm
Arc radii	1,643.2 mm · 2,467.9 mm
Sector angle	87.3°

How to mark it out on the plate

1. Work out g, Ri, Re and θ from the formulas above, or with the calculator, and note the mean circumference of both openings for checking.
2. Fix the trammel centre off the plate, on an offcut or the floor, in line with the edge: the radii are usually longer than the plate (1643 and 2468 mm in the example).
3. Scribe the two arcs: the small one with Ri and the large one with Re.
4. Mark the arc length π × (D2 − 2δ) along the large arc with a flexible tape and scribe the second side from the centre to that point. Both sides are generators and measure g.
5. Check the small arc: it must measure π × (D1 − 2δ) between the two sides. If not, the centre moved between one arc and the other.
6. Cut, roll in crossed passes (a cone does not come off in one pass) and close the seam along a generator.

Common mistakes and tolerances

Entering the slant height as the height

The calculation wants the vertical h. Entering the slant instead gives a taller, tighter cone: in the example, 824.6 in place of 800 gives a sector of 84.8° instead of 87.3°.

Taking the thickness off the radius instead of the normal

Taking t/2 off the radius is right for a cylinder and wrong for a shallow cone. The correct offset is (t/2) × cos α: on an 800 conical cap in 20 mm plate the difference is over 50 mm of arc. On an ordinary thin-wall reducer it is a tenth of a millimetre.

Not taking the thickness off at all

π × D2 on the large arc is the rolled-shell mistake: the large opening comes out π × t long. On 1200 mm in 3 mm plate that is 9.4 mm; in 10 mm plate, 31.4 mm.

Trammel centre moved

Both arcs must come from one centre. If the small arc does not measure π × (D1 − 2δ) between the sides, the centre shifted between arcs.

Minimum taper

The calculation requires D2 − D1 ≥ 1 mm. Below that the sector radii run to infinity and the part is, for all purposes, a rolled shell: use the cylinder page.

Guides about this part

• How to develop a cone (frustum) in sheet metal: formula, sector radii and angle, step by step
• How we check the Planichapa calculations, and who is behind them

Frequently asked questions

Which diameter goes in D1 and which in D2?

D1 is always the smaller and D2 the larger — the calculation rejects the opposite. The height is the vertical distance between the two openings, not the slant length.

How do I mark the sector out without CAD?

With a trammel. The result gives both radii and the sector angle: fix the centre off the plate, scribe the two arcs and close with the straight sides. It is the traditional layout method, with the numbers checked.

Why is the blank slightly smaller than another calculator's?

Because the plate develops on the MEAN surface, and on a cone that sits half a thickness away measured along the surface NORMAL — not along the radius. The reduction on the radius is (t/2) times cos(alpha), smaller than t/2, and smaller still the shallower the cone. Taking off t/2 is right for a cylinder and wrong for a shallow cone: on an 800 Chinese hat in 20 mm plate the difference is over 50 mm of arc. The proof is the limit: at zero height a cone becomes a flat ring, and the blank for a washer is the washer — no reduction at all. On an ordinary thin-wall reducer the difference is a tenth of a millimetre.