Sheet metal
Hopper flat pattern (pyramid frustum)
In short
A rectangle-to-rectangle hopper has four flat trapezoid faces, developed without triangulation. Each face has a true height hf = √(h² + r²), with r the plan setback of the top from the bottom; the lateral edges are k = √(h² + rx² + ry²) and the mass m = S × e × ρ.
Calculation
Example calculated with the tool’s default values.
Inputs
- Bottom opening: width (a1)
- 500 mm
- Bottom opening: depth (b1)
- 350 mm
- Top opening: width (a2)
- 250 mm
- Top opening: depth (b2)
- 180 mm
- Height between the two openings
- 300 mm
- Plate thickness
- 3 mm
- Material
- Carbon steel
- Cutting
- Four separate faces
Results
- Hopper mass
- 9.56 kg3.0 mm, Carbon steel
Worked example
Every value in this section belongs to the example, calculated with the tool’s default values.
Other results of the example
- Developed area of the four faces
- 0.4061 m²
- Longest lateral edge
- 335.9 mm
- Weld length of the 4 edges
- 1343.7 mm
- True height of face AB
- 311.81 mm
- True height of face BC
- 325.00 mm
- Shallowest wall slope
- 67.38°from horizontal (90° = vertical)
- Largest bend angle between faces
- 83.98°
- Dimensions are on the neutral fibre (mid-thickness). For inside or outside dimensions, correct for the thickness before entering them. Bend angles are the press angles: allow for springback and K-factor with your tooling.
Step-by-step calculation
Setback of the top from the bottom (plan)
rAB = b1/2 − (ey + b2/2); rBC = ex − a2/2 + a1/2; (CD and DA likewise)
rAB = 175.0 − 90.0 = 85.00 mm ; rBC = 125.00 mm
True height of a face (distance between its two parallel edges)
hf = √(h² + r²)
AB : √(300.0² + 85.00²) = 311.81 mm ; BC : √(300.0² + 125.00²) = 325.00 mm
Lateral edge (true length)
k = √(h² + rx² + ry²)
A : √(300.0² + 125.00² + 85.00²) = 335.93 mm
Face area (trapezoid)
S = (Lbas + Lhaut) / 2 × hf
AB : (500.0 + 250.0) / 2 × 311.81 = 116928 mm²
Developed area and mass
S = Σ faces ; m = S × e × ρ
406107 mm² × 3.0 mm × 7850 kg/m³ = 9.56 kg
Wall slope
φ = atan2(h, r) (90° = verticale)
AB : atan2(300.0, 85.00) = 74.18° ; BC : 67.38°
Bend angle between adjacent faces
α = angle between the outward normals (inside dihedral = 180° − α)
edge A: 83.98° (dihedral 96.02°); edge B: 83.98°
Pattern of each face (trapezoid)
| Face | Bottom edge (mm) | Top edge (mm) | True height (mm) | Left edge (mm) | Right edge (mm) | Diagonal 1 (check) (mm) | Diagonal 2 (check) (mm) | Area (cm²) | Wall slope (°) |
|---|---|---|---|---|---|---|---|---|---|
| AB | 500.0 | 250.0 | 311.81 | 335.93 | 335.93 | 487.70 | 487.70 | 1169.3 | 74.18 |
| BC | 350.0 | 180.0 | 325.00 | 335.93 | 335.93 | 419.34 | 419.34 | 861.3 | 67.38 |
| CD | 500.0 | 250.0 | 311.81 | 335.93 | 335.93 | 487.70 | 487.70 | 1169.3 | 74.18 |
| DA | 350.0 | 180.0 | 325.00 | 335.93 | 335.93 | 419.34 | 419.34 | 861.3 | 67.38 |
Lateral edges and bend angles
| Edge (corner) | Between faces | True length (mm) | Bend angle (°) | Inside dihedral (°) |
|---|---|---|---|---|
| A | DA / AB | 335.93 | 83.98 | 96.02 |
| B | AB / BC | 335.93 | 83.98 | 96.02 |
| C | BC / CD | 335.93 | 83.98 | 96.02 |
| D | CD / DA | 335.93 | 83.98 | 96.02 |
How to use it
- Enter the bottom opening (width a1, depth b1) and the top opening (a2, b2).
- Enter the height between the two openings.
- Enter the top offsets ex and ey for an offset hopper, or leave them empty if the top is centred.
- Enter the plate thickness and the material.
- Read the hopper mass.
Frequently asked questions
How do I lay out a hopper?
Each face is a trapezoid: its top edge is parallel to its bottom edge. Work out the plan setback of the top from the bottom, r, then the true height of the face hf = √(h² + r²). Draw the bottom edge, the top edge at distance hf, shifted by the setback, and check with the diagonals. Lateral edges are checked at true length: k = √(h² + rx² + ry²).
What does a worked hopper example give?
Example: bottom 500 × 350, top 250 × 180 centred, height 300. Setbacks rAB = 175.0 − 90.0 = 85.00 mm and rBC = 125.00 mm. True heights: AB √(300.0² + 85.00²) = 311.81 mm, BC 325.00 mm. Edge A: √(300.0² + 125.00² + 85.00²) = 335.93 mm. Area of face AB: (500.0 + 250.0) / 2 × 311.81 = 116928 mm².
How are the hopper mass and bend angles calculated?
Mass: m = S × e × ρ, with S the sum of the face areas. In the example, in 3 mm steel: 406107 mm² × 3.0 mm × 7850 kg/m³ = 9.56 kg. The slope of each wall is φ = atan2(h, r): 74.18° for AB and 67.38° for BC. The bend angle between adjacent faces is the angle between their outside normals: 83.98° (inside dihedral 96.02°).
How do I lay out an offset hopper or a vertical wall?
Enter the top offsets ex and ey: the four faces then all differ and must each be developed. A vertical wall needs ex = (a1 − a2) / 2 or ey = (b1 − b2) / 2: its face is then a right trapezoid. If the top overhangs the bottom on one side, that wall overhangs (slope above 90°).
Limits
Indicative results. Dimensions are on the neutral fibre: for inside or outside dimensions, correct for the thickness before entering them. Bend angles are the press angles: allow for springback and K-factor with your tooling.
Sources
- Tronc de pyramide rectangulaire : quatre trapèzes, hauteurs vraies hf = √(h² + r²), arête k = √(h² + rx² + ry²) : Sheetguage, « Truncated Pyramid Flat Pattern Calculator » (consulté le 08/11/2026) (link)
- Réduction rectangulaire (tronc de pyramide) : faces planes, développé exact sans triangulation : Planichapa, « Rectangular Transition development (Pyramid frustum) » (consulté le 08/11/2026) (link)
- Développement par vraies grandeurs : Pattern Development: Sheet Metal Level 1 (Open Textbook BC), chap. 21 « Pyramid » (link)
- Masses volumiques: src/calc/data/materials.ts (EN 10365, EN 10088-1, …)
- Géométrie : Distance de deux points de l’espace, produit vectoriel ; angle de pliage = angle entre les normales extérieures de deux plans