Cone & Frustum Flat Pattern
Develop a cone or frustum into a flat sector: slant height, outer and inner radii, sector angle, arc lengths, chord, blank size and area — with a drawing.
The sector radii and angle to lay out, the two arc lengths, the bounding blank with a seam allowance, the blank area and the half-angle of the cone, plus a scaled sector drawing to check.
Example: A frustum Ø300 to Ø100, 200 high, in 2 mm plate: slant 223.6, outer radius 333.2, inner 109.6, sector 161.0°, on a blank about 667 × 315 with a 10 mm seam.
A cone unrolled
is a sector.
How the sector radii and angle are found, and where the thickness goes.
The development
Cut a right cone along a line from apex to base and unroll it: the surface becomes a sector of a circle whose radius is the slant height and whose arc equals the base circumference. For a frustum the sector is cut short by a second, concentric arc. Slant s = √(((D − d)/2)² + h²); outer radius R₁ = s·D/(D − d) (the full cone’s slant); inner radius R₂ = R₁ − s; sector angle α = 180°·D/R₁ — the angle whose arc at R₁ is exactly π·D.
Thickness and seam
The pattern is drawn for the neutral surface, so the diameters are reduced by one thickness before the arithmetic (outside diameter − T). The seam allowance is added along one straight edge for an overlap or a weld tab and is outside the sector geometry. The blank size is the bounding box of the sector plus that seam.
Layout
From the apex point on the sheet, swing R₁ and R₂ and mark α with a chord: chord = 2·R₁·sin(α/2), which is easier to measure than an angle on a big blank. For thick plate the neutral surface is not exactly mid-thickness; use your shop rule. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Right-cone development: slant s = √(((D − d)/2)² + h²), outer radius R₁ = s·D/(D − d), sector angle 180°·D/R₁, arcs π·D and π·d on the neutral surface (diameters less one thickness)
Last reviewed 20 September 2026. How results are checked: How we verify.