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Triangular Prism Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

60.0000 cm³

Volume

Lateral area
120.0000 cm²
Total surface area
132.0000 cm²

A triangular prism calculator takes the three sides of the triangular end and the length of the prism, and returns three readings: how much it holds, how much wrap goes around its sides, and how much material its whole surface would take. A triangular prism is the shape of a tent, a Toblerone bar, a wedge of cheese, a roof ridge and a length of triangular moulding — anything with a triangle at one end that runs straight along to an identical triangle at the other. Two of the three answers are easy to picture. The volume is the area of the triangular end multiplied by how far the prism runs, because a prism is what you get when you drag a flat shape through space without turning it, and the volume of anything made that way is its cross-section times its travel. The lateral area is the distance around the triangle multiplied by the same length, because unrolled, the sides of the prism are a rectangle whose width is the triangle's perimeter. Put those two together and the third follows: the total surface area is the wrap plus the two triangular ends, one at each end, and that is why the ends are counted twice. The three sides are also asked to be a triangle at all, which is worth a word because it is the check people forget. Three lengths only make a triangle if any two of them add up to more than the third. If they do not, the prism has no end face to start from and no volume to report, and the page says so instead of quietly returning something. When the two shorter sides come to exactly the longest, the end is a flat line rather than a triangle, and the page accepts it: the volume is genuinely zero and the page prints zero.

The volume of a triangular prism from its end and its length

Side a (cm)Side b (cm)Side c (cm)Length (cm)Volume (cm³)
3451060
66610155.8846
556448
234514.5237
112100
34500
2.53.547.532.476
5610222.798

Eight prisms and five columns. The table gives the volume rather than all three readings because five columns is already as wide as this table can carry, and the volume is the one people come for. The first row is the reconciliation row: a 3-4-5 end has an area of exactly 6, so the volume is exactly 60, and the volume page prints the same 60 from a base area of 6 and a length of 10. The second row is the same check with an awkward number — an equilateral end of 6 gives 155.8846, and the volume page reaches that from a base area of 15.5884572. The fifth row is the degenerate end, where 1 + 1 = 2 and the triangle has been flattened into a line: its area is zero, so the volume is zero, and the page accepts it. The sixth row is a length of zero, where the volume is also zero but for an entirely different reason — there is no prism left. The eighth row has an end that is nearly degenerate — 5 + 6 is only just more than 10 — which makes the area tiny and is the hardest case for the arithmetic. Every value here is recomputed from its four inputs when the page is built, in centimetres, and the four decimals are the same four decimals the results panel uses.

Formula

V = A(base) × L

Side a
The first side of the triangular end, in centimetres. The three sides together must be able to form a triangle, which the page checks before answering
Side b
The second side of the end, in centimetres. Together with a and c it fixes the end's area, and therefore the volume, completely
Side c
The third side of the end, in centimetres. The order of the three does not matter — they are used symmetrically, through the perimeter and the semi-perimeter
Length
How far the prism runs, in centimetres — the distance between the two identical triangular ends, measured along the direction of travel
A(base)
The area of the triangular end, worked out from the three sides by Heron's formula. A 3-4-5 end has an area of exactly 6 square centimetres, which is why it appears so often in this group
Volume
How much the prism holds, in cubic centimetres. The end's area multiplied by the length, which works for any prism whatever shape its cross-section is
Perimeter and lateral area
The distance around the end, and that distance multiplied by the length. Unrolled, the sides of any prism are a rectangle, and this is its area

Concrete and earthworks are the oldest users: a triangular berm, a spoil heap with a flat top, a concrete kerb cast in a triangular section — anything ordered or paid for by volume, which is most things sold by the truckload or the cubic metre. Wrap and sheet materials are the second family, and they want the lateral area: the canvas over a tent's ridge, the shrink wrap around a triangular bundle, the cladding on a gable, the sheet metal in a triangular duct. Those jobs are priced by area, and the number they need is the perimeter times the length, not the total surface area, because the ends are usually open or covered by something else. Paint and coating jobs want the total surface area instead, because the ends are part of what has to be painted, and leaving them out of the estimate is the standard way to come up short. Then there is the third reading, which is the one worth knowing about even though it comes free: the volume of any prism is its cross-section area times its length, and once you have seen that, cuboids, cylinders and this shape all collapse into one rule. That generalisation is the reason this page and the volume-of-a-triangular-prism page exist as a pair — one starts from the three sides, the other starts from an area you already have, and both land on the same answer when the numbers line up.

Worked examples

  1. A 3-4-5 end with a length of 10

    1. Work out the area of the end by Heron's formula: semi-perimeter 6, so √(6 × 3 × 2 × 1) = 6
    2. Multiply by the length for the volume: 6 × 10 = 60
    3. Add up the sides and multiply by the length for the wrap: 12 × 10 = 120
    4. Add the two triangular ends: 120 + 2 × 6 = 132

    The pair the page loads with, and the one that ties the pair of pages together: the end has an area of exactly 6, so the volume is exactly 60, and the volume page prints the same 60 for a base area of 6 with a length of 10. All three readings come out of one small set of numbers, which is the point of an umbrella page.

  2. An equilateral end of 6 with a length of 10

    1. Work out the area of the equilateral end: √3 ÷ 4 × 36 = 15.5884572…
    2. Multiply by the length for the volume: 15.5884572… × 10 = 155.884572…

    An equilateral end, where the area is not a whole number and the answer has to be rounded at the end. The volume page reaches the same 155.8846 from a base area of 15.5884572 with the same length, which is the second reconciliation between the two pages.

  3. An isosceles end of 5, 5 and 6 with a length of 4

    1. Work out the area of the end: the semi-perimeter is 8, so √(8 × 3 × 3 × 2) = 12
    2. Multiply by the length for the volume: 12 × 4 = 48

    The ordinary case: an end whose area happens to be a whole number, and a short prism. Nothing here is a special shape and no side had to be measured at a convenient angle, which is what most real jobs look like.

  4. A 3-4-5 end with a length of 0

    1. Work out the area of the end: 6
    2. Multiply by the length: 6 × 0 = 0

    A length of zero means the two ends have been brought together and there is no prism left — the volume is genuinely zero. The surface area is not zero even here, because the two triangular ends are still there and still have to be counted; the lateral area does vanish with the length, since it is the perimeter of the end multiplied by that length. Seeing one of the pair stay put while the other drops to zero is a good way to see that surface area and volume answer different questions.

Limitations

This page takes the three sides of the end and the prism's length, and nothing else. It assumes the prism is right — that the ends are parallel and the same shape, and that the sides run perpendicular to them — which is what the word prism means but not what every triangular solid is; a wedge that tapers toward one end needs different arithmetic. The cross-section must be a triangle: a prism with a general quadrilateral end is a different shape with a different page. The three sides must satisfy the triangle inequality, and when the two shorter ones come to less than the longest the page refuses rather than answering. When they come to exactly the longest, the end is degenerate and the volume is accepted as zero. The answer is always in centimetres, square centimetres and cubic centimetres whatever the dropdowns say, and the conversions are not the same for all three: a length converts by 2.54 to the inch, an area by 6.4516, and a volume by 16.387. Four decimal places is a display width rather than a claim about precision, and a volume with a square root in it will genuinely have more digits than are shown. The lateral area excludes both triangular ends, which is what a wrap material needs but not what a paint estimate needs — for paint, use the total surface area. Nothing here handles a hollow prism, wall thickness, or an allowance for waste when the material is cut.

Frequently asked questions

What is the difference between the lateral area and the surface area?
The lateral area is the wrap around the sides only — the perimeter of the triangle multiplied by the length. The surface area counts the two triangular ends as well, so it is the lateral area plus twice the end's area. Which one you want depends on the job: a wrap, a label or cladding needs the lateral area, because the ends of the bundle are open. Paint, plating or sheet material needs the total, because the ends have to be covered too.
Which measurement is the length?
How far the prism runs — the distance from one triangular end to the other, measured along the direction the shape travels. It is not the height of the triangle, and on a prism lying on a table those two numbers often look similar, which is exactly why it is worth checking which one you have. If your prism is a tent, the length is the ridge.
Can the three sides be any three numbers?
No. Three lengths only form a triangle if any two of them add up to more than the third. If the two shorter ones come to less than the longest, there is no end face and the page refuses rather than answering. If they come to exactly the longest, the end is a flat line and the volume really is zero, so that case is accepted and printed as zero.
Why does the volume page give the same answer for some inputs?
Because the two pages are the same rule reached from different starting points. The volume of any prism is the area of its cross-section times its length, so if you already know the end's area you can skip the three sides entirely. A 3-4-5 end has an area of exactly 6, so both pages give 60 for a length of 10 — one of them working the area out first and the other being told it.
Does it matter which side I call a, b and c?
No. The three sides enter the calculation symmetrically — through the perimeter they are simply added, and through Heron's formula they are used only in the semi-perimeter and the three differences, which are the same set of numbers whichever order you type them in. The labels are there to give the boxes names, not to imply anything.
Can I use this for a cylinder or a cuboid?
Not on this page, because the end here is a triangle and the fields ask for three sides. But the rule is the same one: any prism's volume is its cross-section times its length, and any prism's lateral area is its perimeter times its length. The cuboid page in this group is the same calculation with a rectangular end, and the cylinder page is the same one with a circular end.

References

Related calculators

Cuboid Volume CalculatorVolume and surface area of a rectangular box.Heron's Formula CalculatorTriangle area from three side lengths, by Heron's formula.Surface Area CalculatorSurface area of spheres, cones, cylinders and prisms.Triangle Area CalculatorHalf a base times the perpendicular height above it.