Volume of a Trapezoidal Prism: Formula & Calculator

This calculator works out the volume of a trapezoidal prism from the lengths of the two parallel sides of the trapezoid, the trapezoid’s height, and the prism’s length. Enter those four measurements and you get the volume in cubic units. The formula used is the standard one: volume equals the average of the two parallel sides times the trapezoid height times the prism length.

Formula

The volume of a trapezoidal prism is found by multiplying the area of the trapezoidal cross-section by the length of the prism. In words:

Volume = (1/2) × (side a + side b) × trapezoid height × prism length

In symbols:

V = ½ (a + b) h × L

Where:

• a = length of one parallel side of the trapezoid (base 1), in metres (or any unit)
• b = length of the other parallel side (base 2), same unit
• h = perpendicular height of the trapezoid (distance between the parallel sides), same unit
• L = length of the prism (the distance the trapezoid is extruded), same unit

The result is in cubic units – cubic metres (m³), cubic centimetres (cm³), cubic feet (ft³), etc., matching whatever unit you used for the inputs.

Worked Example

Using the default values from the calculator: side a = 4 m, side b = 6 m, trapezoid height = 3 m, prism length = 10 m.

1. Add the parallel sides: 4 + 6 = 10 m
2. Divide by 2: 10 ÷ 2 = 5 m (this is the average width of the trapezoid)
3. Multiply by the trapezoid height: 5 × 3 = 15 m² (this is the area of the trapezoid face)
4. Multiply by the prism length: 15 × 10 = 150 m³

The volume is 150 cubic metres, which matches the calculator’s output.

Why the Formula Works

A prism is a solid with a constant cross-section. For a trapezoidal prism, that cross-section is a trapezoid. The area of a trapezoid is the average of its two parallel sides multiplied by its height – that’s the ½(a+b)h part. Then, because the prism extends that shape over a length L, you multiply the area by L to get the volume. This is exactly the same logic as for a rectangular prism (length × width × height), except the base area is a trapezoid instead of a rectangle.

You can think of it as slicing the prism into thin slabs of thickness L. Each slab has the same trapezoidal face, so the total volume is just that face area times the total length.

When to Use This Calculator

This calculation appears in construction, landscaping, and engineering. Common real-world uses include:

• Concrete for a trapezoidal footing or channel
• Soil or gravel for a trapezoidal garden bed or ditch
• Water volume in a trapezoidal trough or flume
• Material estimates for a trapezoidal ramp or embankment

If you have a shape that isn’t a perfect prism – for example, a pond with sloping sides that taper – this formula gives an approximation, but for a true prism it is exact.

Common Mistakes and How to Avoid Them

The most frequent error is mixing up the trapezoid’s height with the prism’s length. The trapezoid height is the perpendicular distance between the two parallel sides; the prism length is how far the shape is extruded. Both are needed, but they are different dimensions.

Another common issue is unit inconsistency. If you measure sides in metres but the height in centimetres, the result is off by a factor of 100. Convert all measurements to the same unit before calculating.

Also, be careful with the order of operations. The formula is (a+b)/2 × h × L, not a + (b/2 × h × L). Using the calculator avoids that, but if you do it by hand, use parentheses.

Finally, remember the result is cubic units. If you need litres, 1 m³ = 1000 litres. If you need gallons (US), 1 m³ = 264.172 gallons.

Limitations and Related Shapes

This formula assumes the prism is right – meaning the prism length is perpendicular to the trapezoid face. If the prism is oblique (slanted), the volume is still the cross-section area times the perpendicular length, not the slant length. In practice, most trapezoidal prisms in construction are right prisms.

If the shape is not a prism but a pyramid or a frustum (a trapezoid in 3D that tapers to a point or a smaller shape), this formula does not apply. For a frustum, you need a different formula that accounts for the top and bottom areas.

For a trapezoid itself (2D), the area is ½(a+b)h, which is the first part of this formula. If you only need that, you can ignore the prism length.

Frequently Asked Questions

What is the difference between a trapezoid and a trapezium?

In US English, a trapezoid has at least one pair of parallel sides. In UK English, that shape is called a trapezium, and a trapezoid is a quadrilateral with no parallel sides. This calculator uses the US definition: a shape with one pair of parallel sides.

Can I use this for a trapezoidal pool?

Yes, if the pool has a constant trapezoidal cross-section along its length. Measure the two parallel sides of the cross-section, the vertical height of the trapezoid (the water depth at the shallow and deep ends if it’s a simple trapezoid), and the length of the pool. The result will be the volume. For pools with more complex shapes, break them into simpler prisms.

What if my trapezoid has the parallel sides vertical, not horizontal?

Orientation doesn’t matter. The formula works regardless of which way the trapezoid is rotated. Just identify which two sides are parallel, measure their lengths, and measure the perpendicular distance between them.

How do I convert the volume to litres or gallons?

If you measured in metres, the volume is in cubic metres. Multiply by 1000 to get litres. For US gallons, multiply cubic metres by 264.172. For cubic feet, multiply cubic metres by 35.3147. If you measured in centimetres, the volume is in cubic centimetres; divide by 1000 for litres.

Is the volume the same if I use the average of the sides?

Yes, that’s exactly what the formula does. The average of the two parallel sides (a+b)/2 is the mean width. Multiplying that by the height gives the area of the trapezoid, and then by the length gives the volume. So you can think of it as volume = average width × height × length.

What if the prism is not uniform along its length?

If the cross-section changes along the length, it’s not a true prism. The formula only works for a constant cross-section. For a tapering shape, you would need to integrate or use a different formula for a frustum or other solid.

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in a | Side a | m | 4 | 0
in b | Side b | m | 6 | 0
in h | Trapezoid height | m | 3 | 0
in l | Prism length | m | 10 | 0
out vol | Volume | m³ | (a + b) / 2 * h * l | 2