Quick answer: the volume of any prism is V = base area × length. Find the area of the repeating cross-section, multiply by how far it extends, done. Rectangular and triangular cross-sections have their own dedicated calculators — this page handles everything else.
One rule for every extruded shape
where A = cross-section (base) area · l = length, perpendicular to the cross-section
A prism is any solid you could extrude through a die: the cross-section never changes along its length. That constancy is why the formula is so short — every slice contributes the same area, so the total is just area × length. The only work is finding A.
| Cross-section | Area formula | Example (a or dims) | Area |
|---|---|---|---|
| Rectangle | l × w | 4 × 3 cm | 12 cm² |
| Triangle | ½ b h | b=6, h=4 cm | 12 cm² |
| Trapezoid | ½ (b₁+b₂) h | 4, 2, h=3 cm | 9 cm² |
| Regular pentagon | 1.720 a² | a = 2 cm | 6.88 cm² |
| Regular hexagon | 2.598 a² | a = 2 cm | 10.39 cm² |
| Regular octagon | 4.828 a² | a = 2 cm | 19.31 cm² |
| Circle (→ cylinder) | π r² | r = 2 cm | 12.57 cm² |
For an irregular section — an L-shaped foundation, a stepped machine part — split it into rectangles and triangles, sum the areas, and enter the total here.
Where general prism volume shows up in real life
Hexagonal bar stock
A machinist ordering 1-inch hex brass (a = 0.577 in across each flat-to-corner side) over 36 inches: A = 2.598 × 0.333 = 0.866 in², so V = 31.2 in³. Multiply by brass's density (0.307 lb/in³) and the bar weighs about 9.6 lb — the number that sets the price.
Rain gutters
A K-style gutter's cross-section is about 3.3 in² for a 5-inch gutter. Over a 40-ft run (480 in), that's 1,584 in³ ≈ 6.9 gallons of water when running full — which is why downspout capacity matters more than gutter size in a storm.
L-shaped concrete footings
An L-shaped retaining footing: a 2 × 1 ft slab plus a 1 × 1.5 ft key = 3.5 ft² cross-section. Poured along 60 ft, that's 210 ft³ ≈ 7.8 cubic yards. The concrete calculator converts to truck loads and bags.
Common mistakes & pro tips
- Area in the wrong unit. If the cross-section is in cm² and the length in meters, the raw product is meaningless. This calculator converts per field; on paper, convert first.
- Slant length on a leaning prism. For oblique prisms, use the perpendicular distance between the two end faces, not the longer slanted edge.
- Perimeter instead of area. A hexagon with 2-cm sides has a 12-cm perimeter but a 10.39-cm² area — only the area belongs in the formula.
- Pro tip — weight from volume. Extruded stock is usually bought by weight. Multiply the volume by the material density — the volume to weight calculator has the density tables.
How to use this calculator
- Identify the cross-section. A prism has the same cross-section along its whole length. Find that repeating shape — triangle, hexagon, L-shape, anything.
- Compute the base area. Use the shape's area formula, or for irregular sections, split into rectangles and triangles and add. Enter the result with its square unit.
- Multiply by the length. V = A × l. The calculator converts mismatched units (say, cm² and meters) automatically.
Frequently asked questions
What is the formula for the volume of a prism?
V = base area × length, for every prism. The base can be any flat shape; as long as the cross-section stays constant along the solid, its area times the length gives the volume exactly.
What counts as a prism?
Any solid with the same cross-section from one end to the other: boxes, triangular wedges, hexagonal pencils, I-beams, gutters, extruded aluminum profiles, even a Toblerone bar. If you could push it through a cookie-cutter die, it's a prism.
How do I find the base area of a regular polygon?
Multiply the side length squared by the polygon's constant: pentagon 1.720a², hexagon 2.598a², octagon 4.828a². A hexagonal pencil with 4 mm sides has a cross-section of 2.598 × 16 ≈ 41.6 mm².
Does the prism have to stand upright?
No. An oblique prism — one that leans — holds exactly the same volume as a straight one with the same cross-section and perpendicular length (Cavalieri's principle). Just make sure the length you use is measured perpendicular to the cross-section.