The geometry and its limits

From two circular edges
to one flat pattern.

Coneunwrap develops the side of an ideal truncated cone into a flat paper wrap. The key is to keep both curved-edge lengths while preserving the distance along the side. The result is an annular sector: a portion of a ring.

Three measurements with different jobs

The top and bottom inputs are radii. Measure from the centre of a circular edge to its boundary, or halve its diameter. The slant height is the straight side distance from one edge to the other. It is not the vertical centre-to-centre height. All three measurements must describe the same ideal circular, concentric, straight-sided shape.

Either radius can be the larger one. Coneunwrap labels which edge is wider and uses the larger radius for the developed outer arc. Reversing the top and bottom values changes that label but produces the same flat outline. One radius may be zero for a pointed cone; both cannot be zero.

Why a piece of a ring closes into a cone

Call the larger original radius R, the smaller original radius r, and the slant height s. Their difference is Δ = R − r. When Δ is greater than zero, extend the side lines toward their imaginary cone tip. Similar triangles give the two radii of the developed annular sector:

The difference between those developed radii is exactly s, so each straight seam has the entered side length. The angle is chosen so the outer arc has length 2πR and the inner arc has length 2πr. Those are the circumferences of the original circular edges. The developed radii refer to an imaginary centre on the flat sheet, and can be much larger than the radii of the object.

Worked example: a 90° annular sector

The quarter-ring cut outline occupies a 200 ×200 mm bounding box in this orientation. Its theoretical lateral surface area is π ×(50 +25) ×100 =23,561.944902… mm². The SVG sheet includes additional margins and a calibration area, so its total physical size is larger than the cut-outline bounds. This example is not promised to fit on a single A 4 sheet.

Slant height and vertical height are not interchangeable

A cross-section through the centre forms a right triangle with horizontal leg Δ, vertical leg h and hypotenuse s. Therefore h = √((s − Δ) ×(s + Δ)). For the worked example, h =√9,375 =96.824584… mm. If you entered96.824584 as the slant instead, you would be describing a different surface.

The slant must be strictly greater than Δ. A shorter value cannot form the stated shape. An equal value has zero vertical height and lies flat; that degenerate case is excluded here. A nearly flat valid shape can have a sector angle very close to 360°, but it remains less than a full circle.

Equal radii: the cylinder case

When the two positive radii are exactly equal, Δ is zero. Dividing by it would be meaningless, so Coneunwrap uses the correct cylinder development: a rectangle whose width is 2πR and whose height is s.

For radii that are very close but not identical, the true annular sector has very large developed radii and a very small angle. Coneunwrap keeps that model instead of silently substituting a cylinder. It calculates template points in a nearby local coordinate system so subtracting huge, almost equal radii does not destroy the visible pattern.

What the SVG contains

The SVG contains one closed cut-outline polygon, a 50 mm calibration line, input labels and a clear allowance note. Its width, height and coordinates are expressed in millimetres even if the calculator display uses centimetres or inches. Screen previews deliberately fit the available space and are not a physical ruler.

Curved edges are approximated by short straight segments. The segment count adapts to the geometry so the ideal-circle chord deviation is at most0.02 mm before coordinate rounding. SVG coordinates then round to six decimal millimetres. This is a numerical construction specification, not a claim about printer, cutter or material accuracy. The polygon's curved-edge lengths are slightly shorter than the ideal analytic arcs.

Choose100% or actual-size printing and measure the calibration line after printing. A fit-to-page option can rescale everything. Large patterns may require separate tiling or a larger sheet; automatic page tiling is not part of this tool. Test a paper version before relying on a cut.

Allowances that remain yours to choose

The two straight seams meet edge-to-edge, with zero overlap. There is no glue tab, seam allowance, trimming margin, cutting kerf, thickness compensation, top disc or bottom disc in the cut outline. The SVG's white margin is page space, not an extra part of the shape.

This model treats material as having zero thickness. An outside measurement on a real object may not describe the neutral surface of a thick wrap. It does not account for compression, stretch, adhesive layers or how a joint will be constructed. An oval, offset or curved-sided object generally cannot be represented by these three dimensions.

Units, rounding and checking an export

Use mm, cm or inches consistently. One inch is 25.4 mm. Unit switches retain the canonical millimetre dimensions, while displayed fields round to six decimal places. If you edit a rounded converted value, that new value becomes the measurement. Blanks, negative values, scientific notation and more than six entered decimal places are rejected.

The calculator supports physical radii up to 5,000 mm and slant heights up to 10,000 mm. The bounds limit the inputs, not the paper size: a developed outline can still be very large. Text and CSV exports retain the geometric assumptions and canonical millimetre dimensions. Copying or saving them is a local action; no project is sent to a server.

For an independent check, multiply each developed radius by the sector angle in radians and compare it with the corresponding original circumference. For a cylinder, compare the rectangle width with 2πR. If a reported calculation seems wrong, use Contact with the exact units and inputs. Return to the calculator to develop a new wrap.