Triangular Prism Volume
V = ½ × b × h × L — half the triangle's base times its perpendicular height gives the end area, then multiply by the prism's length. Base 6, height 4, length 10 gives ½ × 6 × 4 × 10 = 120 cubic units. The h is the height of the triangle, not the length of the prism, and not the sloping side.
Where the formula comes from
A triangular prism is the simplest kind of prism: a triangle dragged along a straight line. Because the cross-section never changes, the volume is just cross-sectional area × length.
All the work is in the triangle:
| Step | Working | Result |
|---|---|---|
| Area of the triangular end | ½ × 6 × 4 | 12 square units |
| Multiply by the length | 12 × 10 | 120 cubic units |
The ½ is there because a triangle is exactly half of the rectangle that boxes it in. Draw a 6 × 4 rectangle around the triangle and the triangle fills half of it — no matter where the apex sits along the top.
The height trap
This shape has three different lengths that could all be called "height", and the formula wants only one of them.
| Measurement | Is it the h in the formula? |
|---|---|
| Perpendicular height of the triangle — straight up from the base to the opposite corner | Yes |
| Sloping side of the triangle | No — always longer |
| Length of the prism (how far it runs) | No — that is L |
Mixing up the triangle's height with the prism's length is the most common error here, partly because a prism standing on its end genuinely looks like its "height" runs the other way. The orientation is irrelevant. Find the two identical triangular faces; the distance between them is always L, whichever way the object is sitting.
Any side can be the base. All three sides work, as long as the height you pair with it is perpendicular to that side. Each pairing gives the same area — which makes it a free check on your working if you have the measurements to try two of them.
Equilateral and right-angled triangles
If the triangle is equilateral, do not use the side length as the height. The perpendicular height of an equilateral triangle is only 86.6% of its side:
| Approach | Area | Correct? |
|---|---|---|
| Equilateral formula: (√3/4) × 6² | 15.5885 | Yes |
| Wrongly using side as height: ½ × 6 × 6 | 18 | No — 15% too big |
For a right-angled triangle, life is easier: the two sides meeting at the right angle are the base and the perpendicular height, so ½ × a × b works directly with no extra measuring.
If you only know the three side lengths and no height at all, use Heron's formula: let s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)). That is often the practical route on a real object, where a perpendicular is awkward to measure but the edges are easy.
Reference table
| Base b | Height h | Triangle area | Length L | Volume |
|---|---|---|---|---|
| 6 | 4 | 12 | 10 | 120 |
| 3 | 5 | 7.5 | 12 | 90 |
| 8 | 6 | 24 | 2.5 | 60 |
| 10 | 10 | 50 | 4 | 200 |
| 6 (equilateral) | 5.196 | 15.5885 | 10 | 155.885 |
Where you actually meet this shape
| Object | What is the triangle | What is the length |
|---|---|---|
| Gable roof space / loft | The gable end | The length of the house |
| Wheelchair or loading ramp | The side profile | The width of the ramp |
| V-shaped drainage ditch | The cross-section | The run of the ditch |
| Toblerone box | The end | The length of the bar |
| Tent (ridge type) | The end profile | Front to back |
| Concrete kerb haunching | The triangular fillet | The length of kerb |
A worked case. A gable loft on a house 8 m wide, with a ridge 2.5 m above the ceiling, in a house 12 m long:
- Triangle area = ½ × 8 × 2.5 = 10 m²
- Volume = 10 × 12 = 120 m³
- As air volume for ventilation = 120,000 litres
Note this is the whole roof void. Usable floor area is far less, because you cannot stand near the eaves — a reminder that volume and usable space are different questions.
When it is not a prism
The formula only holds if the cross-section is constant along the whole length. It fails for:
- Hipped roofs — the ends slope inward, so the triangle shrinks toward each end.
- Tapering ditches — the channel narrows or deepens along its run.
- Triangular pyramids — these come to a point, so volume is
⅓ × base × height, notbase × length.
For a shape that tapers uniformly between two different triangular ends, use the frustum approach instead — averaging the two end areas is not correct.
Common mistakes
| Mistake | Effect | Fix |
|---|---|---|
| Using the prism length as the triangle height | Wildly wrong | h belongs to the triangular face; L is the run |
| Using the sloping side as the height | Over-estimates | The height is perpendicular to the base |
| Forgetting the ½ | Exactly double | A triangle is half its bounding rectangle |
| Side as height on an equilateral | 15% too big | Height is 0.866 × side |
| Using ⅓ | You used the pyramid formula | Prisms have no ⅓ |
| Applying it to a hipped roof | Over-estimates | Not a prism — the cross-section changes |
Frequently asked questions
What is the formula for the volume of a triangular prism?
V = ½ × b × h × L — half the triangle's base times its height, times the prism's length. A triangle with base 6 and height 4 has area 12, and over a length of 10 that gives 120 cubic units. The first part is just the area of a triangle; the rest is standard prism logic.
Which height goes in the formula?
The perpendicular height of the triangular face, measured at right angles from the chosen base to the opposite corner. Not the sloping side, and not the length of the prism. Confusing the triangle's height with the prism's length is the most common error here.
How do I find the volume if the triangle is equilateral?
Use (√3/4) × side² for the area, then multiply by the length. A side of 6 gives 15.59, not the 18 you would get by wrongly treating the side as the perpendicular height — an equilateral triangle's height is only about 86.6% of its side.
What if I only know the three side lengths?
Use Heron's formula. Let s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)). Multiply by the prism's length. This avoids needing any perpendicular height, which is useful when measuring a real object where edges are easy but perpendiculars are not.
Is a roof space a triangular prism?
Usually yes for a simple gable roof: the gable end is the cross-section and the length of the house is the prism length. Measure the house width as the base and the rise from ceiling to ridge as the height. Hipped roofs are not prisms — their cross-section changes along the length.
How is a triangular prism different from a triangular pyramid?
A prism has two identical triangular ends joined by rectangles, so cross-section stays constant and volume is area × length. A pyramid tapers to a single point, so volume is ⅓ × base × height. With the same base and height, the pyramid holds exactly one third as much.
Does it matter which side I call the base?
No — as long as the height is perpendicular to that particular side. Any of the three sides can be the base, each with its own matching height, and all three give exactly the same area. That makes it a handy way to check your working.
How do I convert the answer to litres?
From centimetres, divide cm³ by 1000 — a litre is exactly 1000 cm³. From metres, multiply m³ by 1000. A ditch 0.5 m wide, 0.4 m deep and 10 m long has a cross-section of 0.1 m² and a volume of 1 m³, which is 1000 litres.