Cone and Frustum Development Calculator: Flat Pattern Radii and Sector Angle for Hoppers and Transitions
Get the flat pattern for a cone, a hopper or a transition piece.
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Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.
How this calculator works
Enter the large diameter, the small diameter (0 for a full cone) and the vertical height.
The calculator finds the slant height, the radii of the two arcs on the flat pattern, the angle of the sector, the circumferences, the chord to check the layout and the plate area.
Mark out the two arcs from the same centre, to the sector angle, and cut.
Formulas used
- L = \sqrt{H^2 + ((D_1 - D_2)/2)^2}
- R_1 = \frac{L D_1}{D_1 - D_2},\quad R_2 = R_1 - L
- \theta = \frac{180^\circ D_1}{R_1}
Worked example
With the default numbers in the calculator above:
- Slant height of the frustum along the surface: 316.23 mm
- Radius of the large arc on the flat pattern: 632.46 mm
- Radius of the small arc on the flat pattern: 316.23 mm
- Angle of the sector: 113.84 degrees
- Length of the arcs (the circumference of each end): 1256.64 mm and 628.32 mm
- Chord across the large arc (to check the layout): 1059.89 mm
- Surface area of the plate (without laps): 0.30 m2
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
Why is the pattern a part of a circle?
A cone unrolls into a sector. The slant height is its radius, and the circumference of the base is the length of its arc.
What diameters should I use?
Use the diameter at the neutral axis of the plate, about halfway through the thickness, for thick material.
Does it work for an off-centre cone?
No. An eccentric or oblique cone needs a triangulated development.
How do I allow for the seam?
Add the lap or the weld preparation to the pattern after finding the sizes.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any sheet metal pattern development text (Kidder and Parker, Architects' and Builders' Pocket-Book; Machinery's Handbook).
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