How to Find the Volume of a Triangular Prism

A triangular prism is the easiest solid you will meet, once you see what it actually is: one triangle, dragged in a straight line. Every slice across it is the same triangle, the same size and the same shape, so the volume is that one area repeated along the length. There is no second formula to learn and no special case for the shape of the end.

One triangle, dragged along a line

Prism means the cross section never changes. Cut the solid anywhere along its length, square to the ends, and the face you expose is identical to the face you started with. That is the whole reason the volume is simple: you are stacking identical slices, and stacking identical things is just multiplying.

The rule that follows covers every prism, not only triangular ones. Volume is the area of the end times the length. A cylinder is a circle dragged along a line, so its volume is πr² times the length. A box is a rectangle dragged along a line, so it is length times width times height, which is the same rule wearing different clothes. Learn the triangular case properly and you have learned the whole family.

Why it comes out at exactly half a box

Draw the smallest box that will hold the worked prism and you get one measuring 6 by 8 by 15, holding 720. Now take a second copy of the prism, turn it upside down and slide it in beside the first. It fits, with nothing left over. Two prisms fill one box, so each prism holds 360.

That is where the half in half base times height comes from, seen in three dimensions instead of two. It also gives you a sanity check that takes two seconds: sketch the box around the solid, work out its volume, and your answer should be half of it. If you are a long way off half, the mistake is usually in the triangle rather than in the multiplication.

The tidy box only appears when the triangle is right angled. For any other triangle, two copies still fit together, but they make a leaning prism with parallelogram sides rather than a box. The halving survives, the neat picture does not, which is why textbooks reach for right angled ends so often.

A worked check you can repeat on paper

Four routes, one number. The last two rows are the ones worth pausing on. A triangle has three heights, and 4.8 is the one that pairs with the side of 10, so writing 10 times 8 gives 40 rather than 24 and quietly inflates the answer by two thirds. And 408 is the surface area of the very same solid: 408 square units of material to build it, 360 cubic units of space inside it. Two honest answers to two different questions.

When the end is given as three sides

Exam prisms have right angled ends because it makes the area easy. Real ones usually do not, and you are handed three side lengths with no height anywhere. Heron’s formula takes those three sides straight to an area: halve the perimeter to get s, multiply s by s minus each side in turn, then take the square root.

Take a prism 12 long whose ends have sides of 5, 5 and 6. The semi perimeter is 8, so the area is √(8 × 3 × 3 × 2), which is √144, which is 12. The volume is 12 times 12, or 144 cubic units. You can check that end without Heron if you notice it is isosceles: drop a height onto the side of 6 and it splits into two 3, 4, 5 triangles, so the height is 4 and the area is half of 6 times 4, which is 12 again.

Equilateral ends have their own shortcut worth memorising, because they turn up in ridge beams, hoppers and packaging. The area is √3 divided by 4, times the side squared. A prism 20 long with ends of side 10 has an end area of 43.30 and a volume of 866.03. Nothing about that is a whole number, and it should not be. Equilateral triangles carry a square root in their height, so their areas almost never land neatly.

Turning the answer into litres and cubic metres

Cubic units are fine on paper and useless at a builders merchant. Two conversions cover almost everything: a cubic metre is 1000 litres, and a litre is 1000 cubic centimetres. Work in metres from the start if the answer is going to be ordered by the cubic metre, because converting a volume afterwards means cubing the conversion factor and that is where people lose a factor of a million.

A concrete wheelchair ramp is a triangular prism lying on its side. Say it climbs 0.3 m over a run of 3 m and is 1.2 m wide. The end is a right triangle of area 0.45 square metres, so the volume is 0.45 times 1.2, which is 0.54 cubic metres, or 540 litres. Order 0.6 and you have allowed a sensible amount for waste. A ridge tent works the same way in reverse: ends 2.4 m across and 1.5 m tall give 1.8 square metres, and over a 3 m length that is 5.4 cubic metres of air to warm up.

Prism	End area	Length	Volume
Worked example, 6 by 8 by 10 ends	24	15	360
Isosceles ends of 5, 5 and 6	12	12	144
Equilateral ends of side 10	43.30	20	866.03
Concrete ramp, 0.3 rise over 3 run, metres	0.45	1.2	0.54
Ridge tent, 2.4 across and 1.5 tall, metres	1.80	3	5.40

Read down the last column and notice how little the shape of the end matters to the method. Right angled, isosceles, equilateral, a shallow ramp wedge: the second and third columns get multiplied together every single time.

Mistakes that catch people out

• Multiplying all three sides of the triangle. 6 times 8 times 10 is not a step in this calculation and never will be. Only two numbers from the triangle ever meet, a base and the height that belongs to it. The third side sits the whole thing out.
• Stopping at the box. 6 times 8 times 15 is 720, which is the box, not the prism. Halving is not optional, and 720 is the single most common wrong answer.
• Pairing a base with the wrong height. The triangle here has heights of 8, 6 and 4.8, each belonging to a different side. Put the side of 10 with the height of 8 and the end area becomes 40 instead of 24, which is two thirds too big before the length is even applied.
• Using a sloping edge as the length. The length is the distance between the two ends measured square to them. If the prism leans, an edge of 15 tilted 20 degrees off square only spans 14.10, so the true volume is 338 and the 360 you would have written down is 6 percent too much.
• Answering in square units. Three lengths went into the multiplication, so the unit carries a 3. Square units belong to the surface area, which for this prism is a different number entirely.

Try these

Common questions

What is the formula for the volume of a triangular prism?

V = half × base × height × length, which is just the area of the triangular end multiplied by how long the prism is. Ends of 6 by 8 on a prism 15 long give 24 times 15, which is 360 cubic units.

Do you use the third side of the triangle?

Not for the volume. You need one base and the height that goes with it, or all three sides fed through Heron’s formula. The third side does matter for surface area, where every side of the triangle gets a rectangle of its own.

Why is the volume half of length times width times height?

Because two identical triangular prisms fit together to fill a box exactly. The box around the worked prism is 6 by 8 by 15 and holds 720, so one prism holds 360. That halving is the same half that sits in front of base times height.

What if I only know the three sides of the end?

Use Heron’s formula. Halve the perimeter to get s, then take the square root of s times s minus each side. Sides of 5, 5 and 6 give an area of 12, so a prism 12 long holds 144 cubic units.

Is the length of a prism the same as its height?

They are the same measurement when the prism is stood on one of its triangular ends, which is why the words get swapped around so freely. What matters is not the name but the direction: it runs from one triangular end to the other, square to both of them.

How do I turn the answer into litres?

Divide cubic centimetres by 1000, or multiply cubic metres by 1000. A concrete ramp 3 m long, 0.3 m high and 1.2 m wide has a volume of 0.54 cubic metres, which is 540 litres.

Related lessons and calculators

• Surface area of a triangular prism, the same solid answering the other question, and the place the third side finally earns its keep.
• Area of a triangle, since every triangular prism starts and ends with the area of its face.
• Area of a triangle from three sides, Heron’s formula worked through properly for the ends that come without a height.
• Volume of a pyramid, where the same base and the same height give one third instead of the whole.
• Volume of 3D shapes calculator, for prisms, cones and frustums without the arithmetic.
• Area formulas for every common shape, the index this lesson sits inside.
• BIPM on measurement units, the authority behind the cubic metre and the litre.
• Volume of a rectangular prism, the box this prism is half of, and the simplest case of base area times length.
• Volume of a cylinder, the prism rule again, this time with a circular end.