Trapezoid Volume Formula
The formula is V = ½(a + b) × h × L — average the two parallel sides a and b, multiply by the perpendicular height h between them, then multiply by the length L. A flat trapezoid has area, not volume; what you almost certainly want is a trapezoidal prism. British English calls the same shape a trapezium — identical maths.
First: a trapezoid does not have a volume
This sounds pedantic but it resolves most of the confusion around this topic. A trapezoid is flat — a four-sided figure drawn on a page with one pair of parallel sides. Flat shapes have area, measured in square units. Only solids have volume, measured in cubic units.
So when someone searches for "trapezoid volume", they are almost always holding one of these:
- A trapezoidal prism — a trapezoid extruded along a length. A drainage channel, a retaining wall, a loaf tin, a skip.
- A trapezoidal trough — the same thing, open at the top, holding water or grain.
- A frustum — a pyramid or cone with the tip cut off. This looks trapezoid-ish in profile but is not a prism, and needs a different formula. More on that below.
Everything on this page is about the first two. They share one formula.
The trapezoidal prism volume formula
Volume of the prism: V = (a + b) ÷ 2 × h × L
a, b = the two parallel sides · h = perpendicular height between them · L = length of the prism
Read it as three separate ideas and it stops being something to memorise:
- (a + b) ÷ 2 — the average width of the trapezoid.
- × h — average width times height gives the area, exactly as it would for a rectangle.
- × L — area times length gives volume, exactly as it does for every prism.
Step three is worth internalising because it generalises. The volume of any prism, whatever its cross-section, is cross-sectional area × length. Trapezoids are not a special case; they just have a slightly more interesting area formula than a rectangle.
Why averaging the parallel sides is exact
People often assume the average is a convenient approximation. It is not — it is exact, and there is a one-line proof you can do with scissors.
Take your trapezoid. Make an identical copy. Rotate the copy 180° and slide it against the original so the slanted sides meet. The two pieces fit together perfectly into a parallelogram whose base is a + b and whose height is h.
That parallelogram has area (a + b) × h. It is made of two identical trapezoids. So one trapezoid has area (a + b) × h ÷ 2. No approximation anywhere.
The same logic is why the sloped-ceiling averaging trick and the trapezoidal rule in calculus both work — a linear taper always averages exactly.
Worked examples
A drainage channel
A ditch 0.6 m wide at the bottom, 1.4 m across at the top, 0.5 m deep, running 30 m.
- Average width: (0.6 + 1.4) ÷ 2 = 1.0 m
- Cross-sectional area: 1.0 × 0.5 = 0.5 m²
- Volume: 0.5 × 30 = 15 m³ = 15,000 litres
A loaf tin
A tin measuring 8 cm across the base, 10 cm across the top, 7 cm deep and 21 cm long.
- Average width: (8 + 10) ÷ 2 = 9 cm
- Area: 9 × 7 = 63 cm²
- Volume: 63 × 21 = 1,323 cm³ ≈ 1.32 litres
A retaining wall
A concrete wall 0.5 m thick at the base, 0.25 m at the top, 1.8 m tall, running 12 m.
- Average thickness: (0.5 + 0.25) ÷ 2 = 0.375 m
- Cross-section: 0.375 × 1.8 = 0.675 m²
- Volume: 0.675 × 12 = 8.1 m³ of concrete
That last one is the case where getting it wrong costs money — 8.1 m³ is roughly a full truck, and ordering on the base thickness alone would have over-ordered by a third. The concrete volume calculator adds waste allowance on top.
Trapezoid or trapezium? The transatlantic problem
This genuinely confuses people, and the confusion is not their fault — the two words mean opposite things on either side of the Atlantic.
| Shape | American English | British English |
|---|---|---|
| Exactly one pair of parallel sides | Trapezoid | Trapezium |
| No parallel sides | Trapezium | Trapezoid |
For volume purposes it does not matter: the shape with the parallel sides is the one with the formula, whatever your textbook calls it. "Trapezium prism volume formula" and "trapezoidal prism volume" are the same question. If a source uses the word without parallel sides, it has no simple area formula at all and you would need to split it into triangles.
When it is not a prism: frustums and troughs that taper
Here is the mistake that produces genuinely wrong answers rather than just wrong words.
A trapezoidal prism has the same cross-section all the way along. A frustum — a pyramid or cone with the top sliced off — tapers in two directions at once. Its silhouette looks like a trapezoid, so people reach for the prism formula. That is wrong, and the error is large.
| Shape | Cross-section along its length | Formula | Example |
|---|---|---|---|
| Trapezoidal prism | Constant | ½(a+b) × h × L | Drainage ditch, retaining wall |
| Trapezoidal trough | Constant | ½(a+b) × h × L | Feed trough, gutter |
| Truncated pyramid (frustum) | Shrinks in two directions | ⅓h(A₁ + A₂ + √(A₁A₂)) | Skip, planter, hopper |
| Truncated cone | Shrinks, circular | ⅓πh(r₁² + r₁r₂ + r₂²) | Bucket, plant pot |
The test: slice the shape at two different points along its length. If the two slices are identical, it is a prism — use the formula on this page. If the second slice is smaller, it is a frustum — use the truncated pyramid or truncated cone calculator instead.
A builder's skip is the classic trap. It slopes at both ends and both sides, so it is a frustum, not a prism. Treating it as a trapezoidal prism typically overestimates capacity by 10–15%.
Trapezoidal prism volumes — quick reference
Cross-sectional area for common parallel-side pairs, so you can multiply straight by your length.
| a (short side) | b (long side) | Average | h = 0.5 | h = 1.0 | h = 2.0 |
|---|---|---|---|---|---|
| 0.5 | 1.0 | 0.75 | 0.375 | 0.75 | 1.50 |
| 0.6 | 1.4 | 1.00 | 0.500 | 1.00 | 2.00 |
| 1.0 | 2.0 | 1.50 | 0.750 | 1.50 | 3.00 |
| 1.0 | 3.0 | 2.00 | 1.000 | 2.00 | 4.00 |
| 2.0 | 3.0 | 2.50 | 1.250 | 2.50 | 5.00 |
| 2.0 | 4.0 | 3.00 | 1.500 | 3.00 | 6.00 |
| 3.0 | 5.0 | 4.00 | 2.000 | 4.00 | 8.00 |
Units are whatever you use consistently — metres give m², feet give ft². Multiply the area by your prism length for volume.
Common mistakes
- Using a slanted side as the height. Only the perpendicular distance between the parallel sides counts. If you have a slant s and horizontal offset x, then h = √(s² − x²).
- Picking the wrong pair of sides. a and b must be the two that are parallel to each other. On a trough that is the base width and the top width, not the sides.
- Treating a frustum as a prism. The big one. Check whether the cross-section changes along the length.
- Forgetting the ÷ 2. Doubles the answer. If your ditch appears to hold twice what it should, this is why.
- Mixing units. Widths in centimetres and length in metres gives an answer in neither. Convert first.
- Measuring a trough to the rim rather than the fill line. For capacity, b is the width at the liquid surface, not the top of the walls.
Frequently asked questions
What is the volume formula for a trapezoid?
A flat trapezoid has area, not volume — it is a two-dimensional shape. What people almost always mean is a trapezoidal prism, whose volume is V = (a + b) ÷ 2 × h × L, where a and b are the parallel sides, h is the perpendicular distance between them, and L is the length of the prism. In words: average the two parallel sides, multiply by the height, then multiply by the length.
What is the difference between a trapezoid and a trapezium?
They are the same shape under two names. In American English a trapezoid has exactly one pair of parallel sides; in British English that shape is a trapezium. Confusingly the two words also swap meanings — a British trapezoid is a quadrilateral with no parallel sides, which is called a trapezium in American usage. The volume formula is identical either way, so a search for "trapezium prism volume" and "trapezoidal prism volume" will find the same maths.
Why do you average the two parallel sides?
Because a trapezoid is exactly equal in area to a rectangle whose width is the average of its parallel sides. Take two identical trapezoids, rotate one 180°, and slot them together: they form a parallelogram of base (a + b) and height h. That parallelogram has area (a + b) × h, so one trapezoid has half of it. The average is exact, not an approximation.
How do you find the volume of a trapezoidal trough or channel?
Use the same formula, taking a as the width at the bottom, b as the width at the water surface, h as the depth, and L as the length of the channel. A drainage ditch 0.6 m wide at the base, 1.4 m wide at the top, 0.5 m deep and 30 m long holds (0.6 + 1.4) ÷ 2 × 0.5 × 30 = 15 m³, which is 15,000 litres.
Is the slanted side used in the trapezoid volume formula?
No. Only the perpendicular height between the two parallel sides belongs in the formula. The slanted sides are always longer than that perpendicular distance, so using one inflates the answer. If you only know a slant length s and the horizontal offset x, recover the height with h = √(s² − x²).
What is the difference between a trapezoidal prism and a trapezoidal pyramid?
A prism keeps the same trapezoid cross-section along its whole length, so its volume is area × length. A pyramid or frustum tapers, so its cross-section shrinks and its volume carries a factor of ⅓. Confusing them means being wrong by a factor of three. If the shape has a uniform cross-section, it is a prism.