What this calculator does
A trapezium prism is a three-dimensional solid formed by extending a trapezium, a four-sided shape with one pair of parallel sides, along a straight length. Think of a trapezium cross-section stretched out into a beam, a channel or a stockpile of material with sloped sides.
The volume is the trapezium's cross-sectional area multiplied by the length of the prism. This is a genuinely different calculation from the area of a trapezium on its own: the area gives a flat, two-dimensional figure, while this volume adds a third dimension, the prism length, to get a figure in cubic units.
The formula
First find the trapezium's cross-sectional area: half the sum of the two parallel sides, multiplied by the height between them. Then multiply that area by the prism length to get the total volume.
| Term | Meaning |
|---|---|
| a and b | The two parallel sides of the trapezium cross-section. |
| h | The height between the two parallel sides, measured perpendicular to them. |
| L | The prism length: how far the trapezium cross-section is extended. |
| V | Volume, in cubic units. |
The inputs explained
| Field | What to enter |
|---|---|
| Parallel side a | One of the two parallel sides of the trapezium cross-section. |
| Parallel side b | The other parallel side of the trapezium cross-section. |
| Height between the parallel sides | The perpendicular distance between the two parallel sides. |
| Prism length | The length the trapezium cross-section is extended along, at right angles to the cross-section. |
When to use it
A drainage channel or trough with sloped sides
A channel with a wider top than bottom, or vice versa, and a consistent cross-section along its length is a trapezium prism, and this formula gives how much it can hold or how much material was excavated to form it.
A stockpile or embankment with a trapezoidal cross-section
Gravel, soil or aggregate piled with sloped sides and a flat top, or an embankment built to a trapezoidal profile, has a roughly constant cross-section along its length, making this the right volume calculation.
A structural beam or retaining element with a tapered face
A concrete beam, timber member or retaining wall section that is wider at one edge than the other, extended along a fixed length, uses this same trapezium-prism volume.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How volume changes as prism length increases at a fixed cross-section
A fixed trapezium cross-section with parallel sides of 6 and 10 and a height of 4, across a range of prism lengths.
| Prism length | Volume | Cross-sectional area (trapezium) |
|---|---|---|
| 5 | 160.000 | 32.000 |
| 10 | 320.000 | 32.000 |
| 15 | 480.000 | 32.000 |
| 20 | 640.000 | 32.000 |
| 30 | 960.000 | 32.000 |
| 40 | 1,280.00 | 32.000 |
How volume changes as the wider parallel side increases
A fixed narrower side of 6, height of 4 and prism length of 15, across a range of wider parallel sides.
| Parallel side b | Volume | Cross-sectional area (trapezium) |
|---|---|---|
| 6 | 360.000 | 24.000 |
| 8 | 420.000 | 28.000 |
| 10 | 480.000 | 32.000 |
| 14 | 600.000 | 40.000 |
| 20 | 780.000 | 52.000 |
| 30 | 1,080.00 | 72.000 |
Questions
How is this different from the area of a trapezium calculator?
The area of a trapezium calculator gives a flat, two-dimensional area in square units. This calculator takes that same cross-sectional shape and extends it along a length to get a three-dimensional volume in cubic units, which is a genuinely different quantity.
What if the two parallel sides are equal?
A trapezium with equal parallel sides is a rectangle (or parallelogram, depending on the other sides), and the cross-sectional area formula still works, reducing to simply one side times the height.
Does it matter which side is a and which is b?
No. The formula adds the two parallel sides together before halving, so swapping which one is labelled a and which is b makes no difference to the result.
Can this be used for an embankment that tapers at the ends?
This formula assumes a constant cross-section along the full prism length. An embankment or channel that tapers or changes shape at the ends would need to be broken into sections, each treated separately, for an accurate total volume.
For the flat, two-dimensional area of the same trapezium shape, see the area of trapezium calculator. For volumes of other common solids, see the volume of common shapes calculator.