Volume of an arbitrary triangular prism

What is an arbitrary triangular prism?

An arbitrary (general) triangular prism is a geometric solid whose bases are two equal arbitrary triangles connected by three rectangular lateral faces. Unlike a regular triangular prism, where the base is an equilateral triangle, the base of a general prism can be any triangle—scalene, isosceles, or right-angled. It has 6 vertices, 9 edges, and 5 faces.

If the base of your prism is an equilateral triangle, use the regular triangular prism volume calculator—it is simpler to use.

Volume Formulas of a General Triangular Prism

The calculation is performed in two steps:

Step 1. Base Area (Arbitrary Triangle)

The base area is calculated using Heron's formula:

$$s=\frac{a+b+c}{2}$$ $$S_b=\sqrt{s(s-a)(s-b)(s-c)}$$

where a, b, c are the sides of the triangle, and s is the semi-perimeter. To calculate the triangle area separately, use the arbitrary triangle calculator.

Step 2. Prism Volume

The volume of a general right triangular prism is equal to the product of its base area and height:

$$V=S_{b}h$$

where:

• a, b, c—sides of the base (triangle);
• h—height of the prism;
• S_b—area of the base.

Where is the volume of a general triangular prism applied?

• Construction and Earthworks. When designing embankments, dams, trenches, or foundations of a triangular cross-section with uneven sides, volume calculation helps determine the required amount of soil, gravel, sand, or concrete for pouring.
• Ventilation and Air Conditioning. Many air ducts and ventilation shafts in corner areas of rooms or attics have a triangular cross-section with different sides. Calculating the air volume inside such channels is necessary to determine the flow rate and select the appropriate fans.
• Containers and Tanks. Some reservoirs for water or chemicals have a triangular shape for compact placement in the corners of buildings or in truck beds. Knowing the volume helps determine the maximum tank capacity in liters or cubic meters.
• Architecture and Design. Determining the internal volume of rooms under complex roofs (attics, lofts with uneven slopes) is important for designing heating and air conditioning systems.
• Food and Confectionery Production. Calculating the volume of ingredients or finished products of a triangular shape (e.g., cheese wedges, pies, or custom-shaped confectionery products) for correct packaging and dosing.

How to use the calculator?

1. Enter the three base sides a, b, c. If the sides are unknown but you have angles or the height of the triangle, use the arbitrary triangle calculator to find them.
2. Enter the prism height h.
3. Specify the desired precision of the result.
4. Click "Calculate"—the base area and volume will appear instantly.

Frequently Asked Questions (FAQ)

Can a triangular prism have a right-angled triangle as its base?
Yes, this is a very common case. For example, the volume of corner structural elements is often based on a right-angled triangle cross-section. The calculator works with any type of triangle, provided that the triangle inequality condition is met: the sum of any two sides must be greater than the third side.

How to check if three sides form a triangle?
Three segments form a triangle if all three inequalities are satisfied: a + b > c, a + c > b, and b + c > a. If any inequality is not met, the triangle cannot exist, and calculation is impossible. The calculator checks this automatically.

How to convert volume to liters?
1 liter equals 1 cubic decimeter (dm³). If you entered parameters in decimeters, the volume result will immediately be in liters. If in meters, multiply the volume in cubic meters (m³) by 1000 to get liters. If in centimeters, divide the volume in cubic centimeters (cm³) by 1000.