Volume = base area × length. A 3-by-4 rectangular prism 10 long holds 120. A 3-4-5 triangular prism of the same length holds 60, because its base area is 6. The base shape changes, the rule does not.
What makes it a prism
A prism has the same cross-section all the way along. Slice it anywhere perpendicular to its length and you get the identical shape.
That is the whole definition, and it is what makes the volume formula so simple: the cross-section never changes, so the volume is just that area repeated along the length.
It is also why anything tapering is not a prism. A pyramid, a cone or a frustum has a shrinking cross-section, and every one of those carries a factor of a third or worse.
A cylinder is a prism with a circular base, by this reasoning. Most textbooks keep them separate because the circle is a curve, but the volume formula is identical and for the same reason.
Three sides are enough
For a triangular prism, most calculators ask for a base and a perpendicular height. That is awkward, because on a real object the perpendicular height is the one measurement you cannot take directly.
Heron’s formula removes the problem: from the three side lengths alone, area = √(s(s−a)(s−b)(s−c)) where s is half the perimeter.
Three side lengths are exactly what a tape measure gives you. No dropping a perpendicular, no finding a right angle.
It also enforces the triangle inequality for free. If the two shorter sides do not exceed the longest, one of the bracketed terms goes negative and the square root has no real value. This page refuses those inputs and says why, rather than returning a NaN that would propagate silently into the volume.
The lateral surface unrolls
The sides of a prism unroll into a single rectangle, exactly as a cylinder’s wall does.
Its width is the base perimeter and its height is the prism length, so the lateral area is perimeter × length. One formula for every base shape.
It is why a cardboard box net has the four sides in a row: they are one rectangle, scored rather than cut.
The total surface adds both end faces. As with a cylinder, which one you want depends on the object — a duct needs the lateral area, a sealed box needs the total.
Surface to volume
The same ratio that matters for spheres matters here, and prisms make the point in a different way.
A long thin prism has a great deal of surface for its volume. A cube-like one has much less. For a fixed volume, the cube is the rectangular prism with the least surface — the box-shaped analogue of the sphere result.
It is why packaging tends towards cubes when material cost dominates, and why a fin or a heat sink is made of thin plates when the aim is the opposite.
This page reports the ratio alongside the areas, because for anything involving heat, drying or coating it is more useful than either area on its own.
Where it gets used
Construction. Trenches, footings, beams and lintels are prisms, and the volume is what the concrete order is based on.
Ducting and channels. Rectangular ducts and open channels, where the cross-sectional area sets the flow and the perimeter sets the friction.
Packaging. Boxes are rectangular prisms, and the net that folds into one is exactly the lateral rectangle plus two ends.
Optics. A glass prism is a triangular one, and its shape is why it separates light — the refraction differs by wavelength across the two sloping faces.
Sources and methodology
The prism formulas and Heron’s formula are standard; these are the references.
Method. The triangular base uses Heron’s formula so that three measured side lengths are enough, which is what you have when measuring a real object rather than a diagram. Three lengths that cannot close into a triangle are refused with the inequality stated, rather than producing a NaN from the square root of a negative — a failure that would otherwise propagate silently into the volume. That engine is verified on every change against 96 hand-written assertions, including that impossible triangle side lengths are refused with the inequality named rather than returning a NaN, and that a pyramid is exactly a third of the corresponding prism. The count and the per-case breakdown are published on the formula verification page.
Read the guide
The cross-section areas these volumes are built from are on the Quadrilateral Calculator.