BIZOMAT MAKEOS · Geometry
Prismoidal Formula Volume
What is the volume of this prism, pyramid, frustum or earthwork from its end and middle sections?
Volume of any prismoid — prism, pyramid, frustum of a pyramid or cone, and approximately other tapering bodies — from the areas of the two ends and the middle section by the prismoidal formula, with helpers for rectangular and circular ends.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Sections given as | choice | How the sections are described. |
| Height (distance between ends) | mm | Perpendicular distance between the two end sections. |
| Area of end 1 | mm² | Area of one end. |
| Area of the middle section | mm² | Area of the section halfway between the ends (measured, not averaged). |
| Area of end 2 | mm² | Area of the other end. |
| End 1 length (rectangular) | mm | Length of the rectangular end 1. |
| End 1 width (rectangular) | mm | Width of end 1. |
| End 2 length (rectangular) | mm | Length of end 2. |
| End 2 width (rectangular) | mm | Width of end 2. |
| End 1 diameter (circular) | mm | Diameter of circular end 1. |
| End 2 diameter (circular) | mm | Diameter of circular end 2. |
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: Sections given as
- What it is
- How you describe the end sections: as three areas you already know, as two rectangles, or as two circles. Only the inputs for the chosen option are used; the others are ignored.
- Units and format
- Choose one of 3 options from the list: Areas of ends and middle; Rectangular ends (middle averaged); Circular ends (middle diameter averaged).
- Example
- The form opens with “Rectangular ends (middle averaged)” selected as a worked example. Choose the option that matches your case.
- Get it right
- For a body with curved or irregular sides choose areas and measure the middle section; the rectangle and circle options average the end dimensions.
What this means: Height (distance between ends)
- What it is
- The perpendicular distance between the two parallel end sections, measured square to the ends rather than along a sloping side.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 500 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Volume is directly proportional to the height.
- Get it right
- Do not use the slant length of a sloping side; that overstates the volume.
What this means: Area of end 1
- What it is
- The area of the first end section, used only when sections are given as areas. Work it out from the end’s own shape or take it from a CAD section.
- Units and format
- Enter a number in mm²: square millimetres. The form accepts 0 mm² or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 250000 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each unit of end area adds one sixth of the height to the volume.
- Get it right
- Use the same length unit for all areas and the height, so the volume comes out in cubic millimetres.
What this means: Area of the middle section
- What it is
- The area of the section exactly halfway between the two ends, used only when sections are given as areas. Measure or calculate it at that cut.
- Units and format
- Enter a number in mm²: square millimetres. The form accepts 0 mm² or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 140000 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The middle area carries four times the weight of either end, so it dominates the result.
- Get it right
- Measure the real middle section; the average of the two end areas is not the middle area and gives a wrong result.
What this means: Area of end 2
- What it is
- The area of the second end section, parallel to the first, used only when sections are given as areas.
- Units and format
- Enter a number in mm²: square millimetres. The form accepts 0 mm² or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60000 mm² as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Each unit of end area adds one sixth of the height to the volume.
- Get it right
- Either end can be the larger one; the formula is symmetrical.
What this means: End 1 length (rectangular)
- What it is
- The length of the first rectangular end, one side of the rectangle. The tool multiplies it by the end 1 width to get the area and averages it with end 2 for the middle section.
- 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 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Pair lengths with lengths: the side called length on end 1 must run the same way as the length of end 2.
What this means: End 1 width (rectangular)
- What it is
- The width of the first rectangular end, the side at right angles to its 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 400 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure square to the length; a diagonal is not a width.
What this means: End 2 length (rectangular)
- What it is
- The length of the second rectangular end, measured in the same direction as the length of end 1.
- 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
- If the body twists so the sides do not line up, give the sections as areas instead.
What this means: End 2 width (rectangular)
- What it is
- The width of the second rectangular end, measured in the same direction as the width of end 1.
- 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
- A zero here makes end 2 a line, as for a wedge; that is allowed.
What this means: End 1 diameter (circular)
- What it is
- The diameter of the first circular end, across the circle through its centre. The middle section is taken as a circle of the average diameter.
- 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 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the diameter, not the radius; a pointed cone has an end diameter of zero.
What this means: End 2 diameter (circular)
- What it is
- The diameter of the second circular end, parallel to the first, as for the small end of a cone frustum or hopper outlet.
- 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
- Use inside diameters for a container’s capacity and outside diameters for the solid it occupies.
Method and formulas
- V = h (A₁ + 4A_m + A₂) ÷ 6.
- For straight-sided bodies the middle section has the average linear dimensions of the ends; the formula is then exact for prisms, pyramids and frustums.
- Example: rectangular ends 600 × 400 and 300 × 200 mm, 500 mm apart, enclose 70 L.
Limitations
- Exact for bodies whose section area varies as a quadratic along the height; approximate for others.
- Ends must be parallel.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the volume of a hopper, chute or tapered tank
- Estimate the volume of a casting or forging blank between two sections
- Calculate cut-and-fill or stockpile volumes
Who it is for
- Design engineers
- Estimators
- Toolmakers
What you get
- Volume
- End and middle areas used
- Volume in litres
Interpreting the result
- For irregular bodies, measure the middle section rather than averaging the ends.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 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
Flat-sided hoppers and corner angles
What it covers: Develop a square plate hopper: true side shapes, valleys flatter than the sides, the dihedral angle for folding or mitring, thickness at the corners, and a 60° hopper with 50.8° valleys.
Method explained here · BIZOMAT original book
Capacity of a flared tray
In Sheet Metal Production Runs by the Numbers
What it covers: Calculate a flared tray's capacity with the prismoidal formula, compare the shortcuts, size the depth from 200 to 205 mm for 36 L to the fill line, and catch a depth set by average areas.
Method explained here · BIZOMAT original book
Lofts and transitions between sections
What it covers: Loft a round-to-rectangular transition with matched segments, guide curves and end conditions, then check section area and volume along its length before trusting it.
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