The Flat Pattern of a Cone Is a Sector, Not a Circle
Published 9/2/2026 · 6 min read · Everyday calculators
A cone unrolls into a circular sector whose radius is the slant height and whose angle is 360° times the base radius divided by that slant. For a cone of 50 mm base radius and 80 mm height the slant is √(50² + 80²) = 94.34 mm, so the pattern is a 94.34 mm sector spanning 360 × 50 ÷ 94.34 = 190.8 degrees. A truncated cone unrolls into an annular sector: the same construction with the tip cut off, so a frustum of 50 mm and 20 mm radii over 80 mm has an outer pattern radius of 142.40 mm, an inner one of 56.96 and a sector of 126.4 degrees. The angle is never 360 degrees for any real cone — that would be a flat disc.
Roll a 50 mm cone 80 mm tall and its pattern is a 190.8° slice of a 94 mm circle. Flatten a nearly cylindrical lampshade and the pattern radius runs to 839 mm — the shallower the taper, the further away the apex.
Why the angle is 360 times the radius over the slant
The base circle of the cone has to become the outer arc of the sector when the pattern is rolled up, so those two lengths must be equal. The base circumference is 2πr. The arc of a sector of radius s and angle θ is 2πs times θ over 360. Set them equal and everything cancels except θ = 360 r ÷ s. That single line is the entire construction, and it is why the sector angle depends only on the ratio of radius to slant and not on the size of the cone at all.
Two limits fall straight out of it. A very flat cone has a slant barely longer than its radius, so the ratio approaches one and the angle approaches 360 degrees — the pattern becomes a disc with a sliver missing, which at ten millimetres of height is 353 degrees. A very tall cone has a slant far longer than its radius, the ratio goes to zero and the sector narrows to a spike: at 400 mm of height it is 44.7 degrees. Nothing in between is a surprise once the formula is in hand.
A frustum has an apex it never reaches
The pattern of a truncated cone is drawn from the apex of the complete cone it was cut from, even though that apex is nowhere on the object. That is why the outer radius of a frustum's pattern is larger than its slant: for the 50/20 mm frustum over 80 mm, the slant is 85.44 mm but the outer pattern radius is 142.40 — the extra 56.96 mm is the piece that was cut away, and it becomes the inner radius of the annular sector.
The consequence is the one that catches lampshade makers. As the top radius approaches the bottom one, the imaginary apex runs away to infinity and the pattern radius runs with it. Keeping a 100 mm bottom radius and 250 mm of height, the pattern radius is 269 mm for a full cone, 633 mm at a 60 mm top, 1,254 mm at 80 mm, 2,502 mm at 90 mm and 25 metres at 99 mm. A shade you can hold in one hand needs a compass you cannot buy.
How to draw a two-metre arc on a kitchen table
The pattern radius is huge but the pattern itself is not. Take the lampshade: 200 mm across the bottom, 140 across the top, 250 tall. Its slant is 251.8 mm, its outer pattern radius 839 mm and its sector only 42.9 degrees. That sector is a strip 252 mm wide — the difference between the outer and inner radii — and 628 mm long along its outer arc. It fits on a sheet 30 by 70 centimetres, even though the circle it belongs to is 1.68 metres across.
So do not draw the circle. Pin a string of 839 mm to a point on the floor, or use a strip of wood with two holes at 839 and 587 millimetres, and swing only the forty-three degrees you need. Or skip the arc entirely: mark the outer arc's length along a flexible rule laid on the curve and connect the points, which is what the tool's canvas does for you and what a workshop does with a trammel. The geometry is exact either way; only the drawing method has to be practical.
| Height | Slant | Sector angle | Shape of the pattern |
|---|---|---|---|
| 10 mm | 51.0 mm | 353.0° | Almost a full disc |
| 30 mm | 58.3 mm | 308.7° | A disc with a wedge missing |
| 50 mm | 70.7 mm | 254.6° | Two thirds of a disc |
| 80 mm | 94.3 mm | 190.8° | Just over a half-disc |
| 150 mm | 158.1 mm | 113.8° | Under a third |
| 400 mm | 403.1 mm | 44.7° | A narrow wedge |
Frequently asked questions
- How much material does a cone use?
- The lateral area is π times the radius times the slant for a full cone, and π times the sum of the two radii times the slant for a frustum. The 50 mm cone 80 mm tall uses 148.2 cm²; the 50/20 mm frustum over the same height uses 187.9 cm² — more material for less volume, which surprises people until they notice the frustum is wider all the way up. Add a seam allowance along one straight edge and, for a lampshade, a fold at both arcs.
- Can I get the pattern if I only know the slant, not the height?
- Yes, and that is often the more natural measurement to take off an existing object — a tape run up the sloping side gives the slant directly. The tool asks for the height because that is what a drawing usually specifies, but the two are interchangeable through Pythagoras: height = √(slant² − radius²) for a full cone, and √(slant² − (r₁ − r₂)²) for a frustum. Measure whichever you can measure accurately and derive the other.
- Why does my rolled cone not close exactly?
- Almost always because of material thickness. The formula describes a surface with no thickness, so it gives the pattern for the mid-surface of the material; roll a sheet 2 mm thick and the outside travels further than the inside, which shows up as an overlap or a gap at the seam. For sheet metal the trade fix is to compute on the neutral axis — the mid-thickness — and for card or fabric a millimetre of adjustment at the seam absorbs it. Cutting to the mid-surface and allowing a lap joint rather than a butt joint is the practical answer.
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