KEVOS® 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. |
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.
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