Volume Of Triangular Prism Calculator

What this calculator does

A triangular prism is a triangle stretched out into a solid: the two ends are matching triangles, and the sides are rectangles. A Toblerone bar, a ridge tent, a wedge, the roof of a house. This works out its volume from the triangle's base and height, and the prism's length.

Type the three in, pick a unit, and you have the space inside.

Using the calculator

1. Type the length of the prism, how far it runs from one triangular end to the other.
2. Type the triangle's base and its height, the two measurements of the triangular end.
3. Pick the unit, then press Calculate.

All three values have to be positive, and the volume comes back in cubic units of the unit you choose.

The formula | volume = (base × height × length) ÷ 2

The volume of a triangular prism is:

volume = (base × height × length) ÷ 2

The divide by two is the triangle hiding inside it. A triangle's area is ½ × base × height, and that is the area of each triangular end. Multiply that end by the length the prism runs, and you have the volume. So another way to write the same thing is volume = (triangle area) × length.

A triangle carried along its length

This is the base-area-times-length rule that runs through every prism. Picture the triangular end and slide it straight along, sweeping out the solid as it goes. The volume is that end's area times how far it travels. A rectangular prism is a rectangle carried along; a triangular prism is a triangle carried along; a cylinder is a circle carried along. Same idea every time, just a different shape on the end.

And notice what is not here: a divide by three. The cone and pyramid nearby taper to a point, so they lose two thirds and keep a third. A prism keeps the same cross-section the whole way along, so nothing is lost. Its end area carries straight through, in full.

Which measurement is which

The one thing to keep straight is the three roles. The base and the height both belong to the triangular end: the base is one side of that triangle, and the height is the perpendicular distance from that side up to the opposite corner. The length is something else entirely, the distance the prism runs front to back. Swapping the triangle's height for the prism's length is the usual slip, so it is worth a second look before you calculate.

Units and rounding

Because the formula multiplies three lengths overall, the volume comes out in cubic units, matched to the unit you pick: measure in centimetres and the answer is in cubic centimetres, cm³. Keep all three measurements in the same unit before calculating.

A worked example | base 10, height 8, length 12

Say the triangular end has a base of 10 cm and a height of 8 cm, and the prism is 12 cm long.

1. Area of the triangular end: ½ × 10 × 8 = 40 cm².
2. Times the length: 40 × 12 = 480 cm³.

So the volume is 480 cm³. Work out the triangular face first, then carry it the length of the prism, and that is all there is to it.

Questions people ask

What is the volume of a triangular prism with base 10, height 8 and length 12?

It is 480 cm³. Find the triangular end's area, ½ × 10 × 8 = 40, and multiply by the length 12.

Is it the triangle's area times the length?

Yes. The end is a triangle of area ½ × base × height, and multiplying by the length gives the volume.

Why do you divide by 2?

Because the cross-section is a triangle, and a triangle's area is half its base times its height. The divide by two is that half.

Which is the base, the height, and the length?

The base and height are the two measurements of the triangular end. The length is how far the prism runs from one end to the other.

Why no divide by three, like a pyramid?

Because a prism keeps the same triangular cross-section the whole way along. It does not taper to a point, so its end area carries through in full.

References

A note on the principle behind it. The volume of any prism is the area of its cross-section times its length, the base-area-times-height rule shared by all prisms and cylinders. Here the cross-section is a triangle, whose area is one half of its base times its height. For further reading, see Triangular prism.

1. The prism volume principle, the area of the constant cross-section times the length, common to all prisms and cylinders.
2. The triangle area, one half of base times height, giving the area of the prism's two triangular ends.