Cone template generator

How to make a cone or tapered wrap template

1. Measure the object where the wrap starts and ends, not at the lip: the top line, the bottom line and the height between them. A soft tape or a strip of paper round the cup gives the circumference, which is more accurate on a curved cup than a ruler across it.
2. Type the top, the bottom and the height, choose mm, cm or inches, and say whether the height is straight up or along the sloping side. The template and the maths update as you type.
3. Add a seam if you need one: an overlap (the picture runs on into it) for wraps and paper cones, or a plain glue tab for lampshades and card. Add bleed for sublimation, so the picture runs a little past the cut.
4. Open your picture, then move, scale, rotate or repeat it until the turning preview looks right. Tick Mirror if the print goes face-down onto the cup.
5. Download: the PDF to print at 100 % (check the 10 cm line, tape the tiles on the dashed lines), the SVG for a cutting machine, or the PNG at 300 DPI for a sublimation or vinyl printer.

What it cannot do

• It needs your own measurements. There are no cup presets, because the same cup name comes in different sizes; measure yours, and save it here once you have.
• It assumes flat, stiff material. Fabric stretch, paper or card thickness and shrink-wrap film are not modelled, so a thick card cone needs a slightly longer overlap.
• No domes, spheres or other double-curved shapes. Only shapes with straight sides: cylinders, cones and truncated cones. A cup whose side bulges cannot be wrapped flat without creases.
• Screen colours are not print colours. Sublimation inks and home printers shift colours; do a test print.
• Very large templates get a lower DPI. The picture in a PNG or PDF is capped at 16 megapixels, about the most a phone browser can hold, so a big lampshade (884 × 349 mm) comes out near 180 DPI instead of 300; the page says when this happens. The cut lines in the PDF and SVG stay exact.

How the template is worked out

A tapered cup, a lampshade and a party hat are all the side of a cone with its tip cut off (a frustum). Unrolled, that side is a slice of a ring. Call the wider end's diameter D, the narrower end's d and the vertical height h. The side's length is the slant height s = √(h² + ((D − d) ÷ 2)²). The outer edge of the ring has radius R = s × D ÷ (D − d), the inner edge r = R − s, and the slice is θ = 360 × D ÷ (2R) degrees wide, which makes the outer arc exactly π × D long, once round the wide end.

A worked example. A cup 90 mm across the top, 70 mm across the bottom and 200 mm tall: s = √(200² + 10²) = 200.2 mm, R = 200.2 × 90 ÷ 20 = 901.1 mm, r = 700.9 mm and θ = 17.98°. The strip is 281.6 × 208.9 mm, two A4 pages. The smaller the difference between the ends, the bigger R and the gentler the curve; equal ends give a rectangle π × D × h. A party hat with a 150 mm opening and a 200 mm height has d = 0, so r = 0 and the template is a 126.4° slice of a 213.6 mm circle.

The picture is mapped polar: its width goes round the angle and its height down the radius, so a horizontal line in the picture follows one arc, and once wrapped it runs level round the cup. Pictures stretch a little more at the wide end than at the narrow end, because that end is longer; on the cup that is exactly what makes the picture look straight.

Questions

How do I make a tapered tumbler wrap template?

Measure the cup where the wrap will start and end: the diameter (or circumference) at the top line, at the bottom line, and the height between them. Type them in, choose the unit, open your design and download the PNG at 300 DPI for sublimation or the PDF for paper. The strip comes out curved because a tapered cup is part of a cone; a rectangle would wrinkle or gap.

Why is a tapered wrap curved and not a rectangle?

The top and bottom circles of a tapered cup have different lengths, so the wrap has to be a slice of a ring: the wider end is the outer arc and the narrower end the inner arc. For a cup 90 mm across the top, 70 mm across the bottom and 200 mm tall the ring has an outer radius of 901 mm and an angle of 17.98°, a gentle curve. Only when both ends are equal is the wrap a rectangle, π × D wide.

Can I put my own picture on the cone template?

Yes. Open a picture and it is bent onto the strip: across the picture becomes round the cup and down the picture becomes down the side, so a horizontal line on the picture lands on one arc and comes out level on the cup. Move, scale, rotate and repeat it round the cup, and switch on Mirror for transfers that are printed face-down. The picture is read on your device and never uploaded.

How do I make a lampshade or paper cone template bigger than my paper?

The PDF tiles the strip over as many A4 or Letter pages as it needs, with 10 mm dashed overlap lines and a label on every tile (A1, A2, B1 and so on). A lampshade 200 mm across the top, 300 mm across the bottom and 250 mm tall is an 884 × 349 mm strip, which takes 10 A4 pages or 8 Letter pages. Every page has a 10 cm check line so you can see the printer did not scale it.

Can I cut the template on a cutting machine?

Yes. The SVG is drawn in millimetres with the two edges as true circular arcs (radius R and r), so it imports at the right size into cutting-machine and laser software. The seam line is a separate red dashed path you can set to score or hide. Turn on Mirror before downloading if you cut heat-transfer vinyl face-down.