BIZOMAT MAKEOS · Geometry
Hopper and Frustum Volume
What volume and side area does this tapered hopper or frustum have?
Volume and slant surface area of a round (cone frustum) or rectangular (pyramid frustum) hopper, transition or bin.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Shape | choice | Round for cones, rectangular for pyramid-shaped hoppers. |
| Top diameter or length | mm | Inside diameter (round) or length (rectangular) at the top. |
| Top width (rectangular only) | mm | Inside width at the top; ignored for round. |
| Bottom diameter or length | mm | Inside diameter or length at the outlet; 0 for a full cone or pyramid. |
| Bottom width (rectangular only) | mm | Inside width at the outlet; ignored for round. |
| Vertical height | mm | Vertical distance between the top and bottom planes. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Shape
- What it is
- Whether the hopper or transition is round (a cone with its tip cut off) or rectangular (a pyramid with its tip cut off).
- Units and format
- Choose one of 2 options from the list: Round (cone frustum); Rectangular (pyramid frustum).
- Example
- The form opens with “Round (cone frustum)” selected as a worked example. Choose the option that matches your case.
- Get it right
- For a round shape the two width inputs are ignored.
What this means: Top diameter or length
- What it is
- At the wide top opening: the inside diameter for a round hopper, or the inside length of one side for a rectangular hopper.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1000 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use inside dimensions for capacity; outside dimensions overstate the volume by the plate thickness.
What this means: Top width (rectangular only)
- What it is
- The inside width of the top opening of a rectangular hopper, measured at right angles to the top length.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 800 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored for a round hopper.
What this means: Bottom diameter or length
- What it is
- At the small bottom outlet: the inside diameter for a round hopper, or the inside length for a rectangular hopper, on the same side as the top length.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Set it to 0 for a full cone or pyramid that comes to a point.
What this means: Bottom width (rectangular only)
- What it is
- The inside width of the bottom outlet of a rectangular hopper, on the same side as the top width.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored for a round hopper. Match it to the top width, not the top length, or the sides will be wrong.
What this means: Vertical height
- What it is
- The vertical distance between the plane of the top opening and the plane of the outlet, not the sloping length of a side.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 900 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Volume rises in proportion to the height for the same openings.
- Get it right
- Measure square to the openings; the slant height is calculated from this.
Method and formulas
- Round: V = πh(D² + Dd + d²) ÷ 12; side area = π(R + r) × √((R − r)² + h²).
- Rectangular: V = h(A₁ + A₂ + 4Aₘ) ÷ 6 (prismoidal formula, exact for a frustum), with Aₘ the mid-height section.
- Rectangular side area = sum of four trapezoids with slant heights √(h² + ((top − bottom) ÷ 2)²).
- Example: a round hopper 1000 mm to 300 mm over 900 mm holds 327.51 L with 1.9719 m² of side area.
Limitations
- Centred (symmetrical) frustum assumed; offset outlets change the side area.
- Plate thickness, laps and seams are not included in the area.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size hoppers and chutes
- Estimate sheet area for a fabricated transition
- Check bin capacity
Who it is for
- Fabrication estimators
- Design engineers
What you get
- Volume
- Slant surface area
- Slant height and side angle
Interpreting the result
- Check the side angle against the material flow requirement for the stored product.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 82–87, 66. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Installing a weighing vessel
In Strain Gauges and Load Cells by the Numbers
What it covers: How to size a hopper's load cells from its volume and load sharing, mount them so the vessel can grow and move freely, and stop pipes, chutes and cables from carrying weight past the cells.
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