How this instrument works
A trapezoidal prism is what results from taking a trapezoid — two parallel sides, called bases, separated by a perpendicular height — and dragging it in a straight line for some distance. The cross-section never changes shape as it travels, so its area stays fixed at ½(b₁ + b₂)h, the same trapezoid-area relationship worked out on its own elsewhere on this site; this page's only new idea is what happens once that flat figure is given a third dimension to move through.
Multiplying a fixed cross-sectional area by the distance it travels is the rule for any prism's volume, whatever shape the cross-section happens to be — triangular, trapezoidal, hexagonal, or an irregular outline traced from a survey. A canal, a roof gutter, a drainage swale, and a poured concrete retaining wall are all trapezoidal in cross-section far more often than rectangular, because sloped sides resist collapse and shed water better than vertical ones; this is the formula engineers actually reach for when pricing the earth moved or the concrete poured along such a run.
A genuinely surprising consequence follows: two trapezoidal prisms with entirely different base measurements hold identical volume the instant their cross-sectional areas match, regardless of how the individual sides b₁, b₂, and h happen to be split. And pushed to an edge, the formula degrades gracefully rather than breaking — collapse one base to zero and the cross-section becomes a triangle, so the volume formula quietly turns into a triangular prism's V = ½bh × length without any special-casing anywhere in the arithmetic.
- Enter the base trapezoid's two parallel sides into Base trapezoid: side 1 and Base trapezoid: side 2 — either order works, since addition doesn't care which is longer.
- Enter the perpendicular gap between those two sides into Base trapezoid: height, not the length of a slanted leg.
- Enter how far that trapezoid is extruded into Prism length, using the same unit as the three base measurements.
- Read the result in Volume, reported in that shared unit cubed.
Worked example — a 6-and-10 trapezoid extruded 10 units
A drainage channel's cross-section is a trapezoid with parallel sides of 6 and 10 units and a height of 4 units, extruded along a run of 10 units: Base trapezoid: side 1 = 6, Base trapezoid: side 2 = 10, Base trapezoid: height = 4, Prism length = 10. The cross-sectional area comes first, A = ½(6 + 10) × 4 = ½ × 16 × 4 = 32 square units, and Volume follows in one more multiplication: V = 32 × 10 = 320 cubic units.
Wall off one bank of that same channel — drop Base trapezoid: side 1 to 0 and leave the other three fields untouched — and the trapezoid degenerates into a triangle of area ½ × 10 × 4 = 20 square units, so Volume drops in step to 200 cubic units over the same 10-unit run. The formula never needed to be told it was now looking at a triangular prism instead, because a triangle is simply a trapezoid with one base collapsed to zero.
Questions
What is the formula for the volume of a trapezoidal prism?
V = ½(b₁ + b₂)h × length: first find the base trapezoid's own area with ½(b₁ + b₂)h, then multiply that area straight through by how far the shape is extruded. With sides 6 and 10, a height of 4, and a length of 10, the area works out to 32 and the volume to 32 × 10 = 320.
How does trapezoidal prism volume relate to a trapezoid's area?
It's that same flat area multiplied by one more dimension. The ½(b₁ + b₂)h term inside the volume formula is exactly the base trapezoid's own area — nothing about it changes when the shape is extruded — so anything already known about that area carries straight over before the final multiplication by length.
How is this different from the lateral surface area of a trapezoidal prism?
Volume measures the space enclosed inside the solid, in cubic units, using the base's parallel sides and its height. Lateral surface area measures the material needed to wrap the four flat side walls, in square units, using only the base's perimeter and length — a related but separate question answered on its own page.
Why does a Base trapezoid: side value of 0 still give a valid answer?
Because collapsing one parallel side to zero turns the trapezoidal cross-section into a triangle, and the formula keeps working without complaint: with side 1 at 0, side 2 at 10, height 4, and length 10, the cross-section's area drops to 20 and the volume to 200 — exactly what a triangular prism's own volume formula would return.
Does this volume formula work for any cross-section shape, not just trapezoids?
The underlying principle does — area of the cross-section times the distance it's extruded — but the ½(b₁ + b₂)h term itself is specific to a trapezoidal base. Swap in a circle's area for a cylinder, a triangle's for a triangular prism, or a rectangle's for a cuboid, and the same area-times-length logic still supplies the volume.
What's a common mistake when measuring Base trapezoid: height?
Entering a leg's slanted length instead of the perpendicular height between the two parallel sides — the same mix-up that trips people on the plain 2D trapezoid-area page. Only in a right trapezoid does one leg's length happen to equal the height; everywhere else, measure straight across, not along the slope.