Half differences
ΔL = (Lt − Lb) / 2
ΔW = (Wt − Wb) / 2
These are the horizontal offsets between the centre lines of the top and outlet rectangles.
Existing engineering calculator
Hopper Valley Angle Calculator
Calculate a rectangular hopper’s valley angle or height, along with the two plate angles, surface area and internal volume.
Calculator inputs and result
Choose whether to calculate valley angle from hopper height or height from valley angle. Enter all opening dimensions in millimetres.
Engineering knowledge
Hopper plate geometry, valley angle and frustum volume
A rectangular hopper transitions from a larger top opening to a smaller outlet using four inclined fabricated plates. The valley angle describes the inclination along the line where two adjacent plates meet. It is useful for preliminary hopper layouts, plate-development checks, material estimates and early equipment sizing.
Related calculation knowledge
Hopper geometry alone does not establish reliable bulk-solid flow. Material properties, wall friction, outlet size, discharge arrangement and structural loads require separate assessment.
Explore Material Handling and Bulk Solids →Input and result definitions
Top and outlet openingsLt and Wt are the top length and width; Lb and Wb are the corresponding smaller outlet dimensions.
Calculation modeEnter height to calculate the valley angle, or enter valley angle to calculate height.
Reported geometryThe result gives the two plate angles plus total surface area and internal frustum volume.
Formula and calculation method
Valley relation
θᵥ = tan⁻¹[H / √(ΔL² + ΔW²)]
When valley angle is entered instead, the existing calculator rearranges this relation to determine H.
Frustum volume
V = H / 3 [A₁ + A₂ + √(A₁A₂)]
A₁ and A₂ are the top and outlet plan areas, calculated in consistent metric units.
Assumptions and limitations
- The existing relation assumes the outlet is centred and each wall is a flat plane between rectangular openings.
- Surface area represents the primary hopper plates; it does not include flanges, stiffeners, liners, outlet spigots, weld allowances or access items.
- The result is a geometric estimate only. It does not determine mass-flow performance, outlet sizing, wall thickness or structural adequacy.
- For actual bulk-solid service, evaluate the stored material, moisture, wall friction, discharge equipment, dynamic loads and project-specific standards.
Illustrative use case
For a 3,000 mm by 2,500 mm top opening, a centred 600 mm by 600 mm outlet and a 2,200 mm hopper height, the existing calculator reports the two plate angles, valley angle, total plate surface area and internal frustum volume. These values can support a preliminary layout before detailed flow and structural design.
Frequently asked questions
What is the difference between a plate angle and a valley angle?
A plate angle is measured along one hopper wall. The valley angle is measured along the seam where two adjacent inclined plates meet, so it accounts for the offsets in both directions.
Can this calculator be used for square hoppers?
Yes. Enter equal top length and width, and equal outlet length and width, for a square hopper. The calculator also accepts rectangular openings.
Does the result confirm that material will flow?
No. Geometry by itself cannot predict reliable flow. Bulk material testing and a hopper-flow assessment are required for a design decision.
Can I use the surface area for plate weight?
It provides a preliminary main-plate area. Add allowances and account for thickness, density, cutouts, stiffeners, welds and lining separately before procurement or fabrication.
References
- Jenike, A. W. Storage and Flow of Solids. University of Utah Engineering Experiment Station Bulletin No. 123. 1964.
- McGlinchey, D. Bulk Solids Handling: Equipment Selection and Operation. Blackwell Publishing. 2008.
- Oberg, E., Jones, F. D., Horton, H. L. and Ryffel, H. H. Machinery's Handbook. 30th ed. Industrial Press. 2016.
This is an original educational summary. Verify material data and project requirements for actual work.
Review information
Content type: existing engineering calculator with knowledge module.Interface reviewed: 27 August 2026. Technical review is required before project use, procurement, construction, operation, compliance or safety decisions.
Engineering disclaimer
Educational and preliminary engineering-reference use only.This calculator does not replace bulk-solid testing, project specifications, detailed design, fabrication drawings, applicable standards or review by a qualified engineer. Verify all values, assumptions and decisions for actual service conditions.