How this calculator works
The triangular end is the cross section of the prism, so the volume is that end area times the length. Heron's formula computes the end area from the three side lengths alone, with no need for a height. The three rectangular sides together have a total width equal to the triangle perimeter, so their area is the perimeter times the prism length. Adding the two ends gives the total surface area. In solve mode, dividing a known volume by the end area recovers the length.
Formula
s = (a + b + c)/2; end area = √(s(s-a)(s-b)(s-c)); volume = end area × length; lateral area = (a + b + c) × length; total area = 2 × end area + lateral area.
Worked example
A prism with a 3-4-5 triangular end and a length of 10 m has an end area of 6 m², a volume of 60 m³ and a lateral area of 12 × 10 = 120 m². The total surface area is 2 × 6 + 120 = 132 m². If the volume is 30 m³ instead, the length comes back as 5 m.
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Frequently asked questions
What kind of triangle can I enter?
Any triangle that satisfies the triangle inequality. Right, equilateral, isosceles and scalene triangles all work. If one side is longer than the other two combined, no triangle exists and the input is rejected.
Why does Heron's formula work without a height?
The three side lengths already fix the shape of a triangle completely. Heron's formula converts those lengths directly into area, so a separately measured height is unnecessary.
Can I find the prism length from a volume?
Yes. Choose the volume mode, enter the volume and the three triangle sides, and the calculator divides the volume by the end area to give the length.