Paper Cup Taper and Nesting: Why Six Degrees Decides Your Carton Count
Two cups that look identical can nest 8 mm apart or 4 mm apart, and that 4 mm decides how many cups fit in a carton, how high a dispenser stack stands, and whether the cups jam in a vending machine. The controlling number is the taper angle of the cup wall, usually 6 to 8 degrees, and it interacts with wall thickness and rim diameter in ways that are easy to get wrong.
Why Cups Nest at All
A paper cup is a truncated cone. When one cup enters another, it stops when the outer surface of the inner cup touches the inner surface of the outer cup. The distance it travels depends on the diameter difference per unit height, which is the taper, and on the wall thickness, which sets the initial clearance.
A cup with 6 degrees of taper and a 0.35 mm wall advances about 7 to 9 mm per nesting step. At 8 degrees the step grows to 10 to 12 mm. At 4 degrees it falls to 4 to 5 mm and the cups begin to bind.
The Six Degree Number
Six degrees is the industry default because it balances two failures. Too little taper and the cups wedge: extraction force climbs above 25 newtons and a dispenser cannot separate them. Too much taper and the stack is loose: cups wobble, the stack tilts in the carton, and the rim of one cup can ride over the rim of another and lock.
Measure the taper as the included half-angle from the vertical, not the full cone angle. A "12 degree cone" is 6 degrees per side, and suppliers quote both, which is a reliable source of confusion in drawings.
The Nesting Arithmetic
For a 90 mm rim diameter cup with a 6 degree half-angle and a 0.35 mm wall, the nesting step is about 8 mm. Fifty cups therefore form a stack of 400 mm plus the first cup height of 110 mm, or 510 mm. At a 5 mm step, the same 50 cups reach 360 mm, and at 10 mm they reach 610 mm.
Carton count follows directly. A 600 mm carton holds 60 cups at an 8 mm step, 75 at 5 mm and 48 at 10 mm. That is a 25 percent swing in units per carton from a taper change nobody notices on a drawing.
Wall Thickness and Rim Interference
Wall thickness of 0.30 to 0.40 mm is standard for a 250 ml hot cup. At 0.28 mm the cup is 15 to 20 percent weaker in hoop compression and deforms when gripped; at 0.45 mm the nesting step grows by 1.5 to 2 mm and the carton count falls by 8 to 10 percent.
The rim is the other interference point. A rolled rim of 2.4 to 2.8 mm diameter must clear the inner wall of the cup below. If the rim diameter tolerance is plus or minus 0.3 mm and the wall is at the top of its tolerance, the rim contacts the wall and the cup stops early, producing a short stack and a jam at the denester.
Too Little Taper: The Jam
Below 4.5 degrees, the contact area between cups becomes large and friction dominates. Extraction force rises from 8 to 12 newtons at 6 degrees to 30 to 45 newtons at 4 degrees, and a dispenser set for 20 newtons will crush the cup instead of releasing it.
Humidity makes it worse. At 80 percent relative humidity the paper wall swells 0.02 to 0.04 mm, which is enough to close the clearance on a low-taper cup. If the product ships to a humid market, keep taper at 6 degrees or above.
Too Much Taper: The Wobble
Above 9 degrees the stack is unstable. Cups tilt by 2 to 4 degrees from the stack axis, cartons bulge, and the rim of a tilted cup can catch the rim below, creating a double-cup that the dispenser serves as one. Double-cupping rates above 2 percent are almost always a taper or rim tolerance problem.
The visual symptom is a stack that leans in the sleeve. The fix is not a tighter sleeve; it is a taper reduction to 7 degrees and a rim diameter control of plus or minus 0.15 mm.
Specification
Write: taper 6.5 degrees plus or minus 0.3 per side, wall 0.35 mm plus or minus 0.03, rim diameter 90.0 mm plus or minus 0.15, nesting step 7.5 to 9.0 mm, extraction force 6 to 14 newtons, carton count 60.
Then measure 10 cups from each lot with a caliper and a stack-height gauge. Taper is set by the forming mandrel and does not drift often, but wall thickness and rim diameter do, and those two are what turn a good design into a jammed line.