Volume of a Trapezoidal Prism Calculator
Enter the two parallel base lengths, the perpendicular height of the trapezoid, the two non-parallel leg lengths, and the length of the prism to get the volume, total surface area, and lateral surface area in one step. Switch between metric and imperial units at any time. A worked-out panel shows every step of the arithmetic so you can follow exactly how each result was reached.
Formula
Worked example
A trapezoidal prism has b1 = 8 m, b2 = 4 m, trapezoid height h = 3 m, both legs = 3.606 m, and prism length L = 10 m. Base area = (1/2)(8+4)(3) = 18 m². Volume = 18 x 10 = 180 m³. Perimeter = 8 + 4 + 3.606 + 3.606 = 19.212 m. Lateral area = 19.212 x 10 = 192.12 m². Total surface area = 192.12 + 2 x 18 = 228.12 m².
What is a trapezoidal prism?
A trapezoidal prism is a three-dimensional solid whose two congruent, parallel faces are trapezoids and whose remaining four faces are rectangles. It has 6 faces, 12 edges, and 8 vertices. The shape is common in civil and structural engineering: irrigation canals, road embankments, dams, and retaining walls all have cross-sections that are trapezoidal, so their volumes are computed with this formula. In everyday terms, think of a wedge-shaped trough that is wider at the bottom than the top, or a concrete beam that tapers slightly from one face to the other.
How to calculate the volume
The volume of a trapezoidal prism equals the area of the trapezoidal base multiplied by the length of the prism. The base area of any trapezoid is half the sum of the two parallel sides (the bases) times the perpendicular height between them: A = (1/2)(b1 + b2) x h. Multiplying by the prism length L gives V = (1/2)(b1 + b2) x h x L. For example, a prism with b1 = 8 m, b2 = 4 m, h = 3 m, and L = 10 m has base area = (1/2)(12)(3) = 18 m², so volume = 18 x 10 = 180 m³. The "Show your work" panel in this calculator walks through every arithmetic step with your own numbers.
Surface area and lateral area
The total surface area includes both trapezoidal bases and the four rectangular faces that connect them. The four lateral rectangles together form the lateral surface area, calculated as the perimeter of the trapezoidal base times the prism length: LSA = (b1 + b2 + a + c) x L, where a and c are the lengths of the two non-parallel legs of the trapezoid. The total surface area is then TSA = 2 x A_base + LSA. For an isoscopic trapezoid (both legs equal) you only need one leg value. If you do not enter the leg lengths this calculator still gives you the volume and base area; the surface area fields stay blank until both legs are provided.
Real-world uses and unit tips
Volume of a trapezoidal prism appears in earthworks (cut-and-fill calculations for roads, canals, and embankments), civil engineering (estimating concrete volumes for trapezoidal foundations and retaining walls), and fluid storage (open channels and flumes). When working in metres, the result is in cubic metres (1 m³ = 1,000 litres). Switch to centimetres and the result is in cm³ (1 cm³ = 1 mL). In imperial units, cubic feet can be converted to US gallons by multiplying by 7.4805. Always confirm that all six dimension inputs use the same length unit before calculating.
Common trapezoidal prism applications and typical dimensions
| Application | Typical b1 | Typical b2 | Height h | Length L | Approx. volume |
|---|---|---|---|---|---|
| Irrigation canal (small) | 3 m | 1.5 m | 1.2 m | 100 m | 270 m³ |
| Road embankment cross-section | 20 m | 10 m | 4 m | 50 m | 3,000 m³ |
| Concrete retaining wall (trapezoidal base) | 0.8 m | 0.3 m | 1.5 m | 5 m | 4.1 m³ |
| Flower bed / raised planter (keystone shape) | 1.2 m | 0.6 m | 0.3 m | 2 m | 0.54 m³ |
| Dam cross-section (simplified) | 60 m | 8 m | 40 m | 200 m | 272,000 m³ |
Approximate dimensions and volumes for everyday trapezoidal prism shapes.
Frequently asked questions
What is the formula for the volume of a trapezoidal prism?
V = (1/2) x (b1 + b2) x h x L, where b1 and b2 are the two parallel sides of the trapezoidal cross-section, h is the perpendicular height of the trapezoid, and L is the length of the prism. Equivalently, V = base area x L, where base area = (1/2)(b1 + b2)h.
What is the difference between the prism length and the trapezoid height?
The trapezoid height (h) is the perpendicular distance between the two parallel sides within the trapezoidal cross-section - it is measured across the face of the trapezoid. The prism length (L) is the distance between the two trapezoidal faces - essentially the depth or extrusion length of the solid. Both are needed to calculate volume.
Do I need the leg lengths to calculate volume?
No. Volume depends only on the two parallel base lengths, the trapezoid height, and the prism length. The leg lengths (the two non-parallel sides of the trapezoid) are only needed to compute the perimeter, lateral surface area, and total surface area. You can leave the leg fields at zero to get just the volume and base area.
How do I find the leg lengths if I do not know them?
For a right trapezoid (one vertical leg), one leg equals the trapezoid height h and the other can be found with the Pythagorean theorem: c = sqrt(h^2 + (b1-b2)^2). For an isosceles trapezoid (symmetrical, equal legs), each leg is sqrt(h^2 + ((b1-b2)/2)^2). If the prism is a right-angled channel with a vertical side and a sloped side, you need to know both independently.
How is the lateral surface area of a trapezoidal prism calculated?
The lateral surface area (LSA) is the total area of the four rectangular faces that run along the length of the prism. It equals the perimeter of the trapezoidal base times the prism length: LSA = (b1 + b2 + leg1 + leg2) x L. Adding the areas of the two trapezoidal bases (each equal to (1/2)(b1+b2)h) gives the total surface area: TSA = LSA + 2 x base area.
What units does this calculator use?
You can choose metric (metres, centimetres, or millimetres) or imperial (feet or inches). All dimension inputs must use the same unit. Volume is then reported in the corresponding cubic unit (m³, cm³, mm³, ft³, or in³) and surface areas in the corresponding square unit. To convert m³ to litres, multiply by 1,000. To convert ft³ to US gallons, multiply by 7.4805.
Can volume be a negative number?
No. Volume is always a non-negative quantity. All dimension inputs must be positive numbers. If any dimension is zero or negative the calculator leaves the result blank and waits for valid input.