Triangular Prism Calculator

Calculate volume, surface area, and more for any triangular prism.

Enter the lengths of the three sides of the triangular base (a, b, c) and the height of the prism (h). The calculator will determine the volume and surface areas.

Examples

Click an example to load its data into the calculator.

Right-Angled Triangle Base

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Calculate properties of a prism with a common 3-4-5 right-angled triangle base.

a: 3, b: 4, c: 5

h: 10

Equilateral Triangle Base

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Calculate properties of a prism where the base is an equilateral triangle.

a: 6, b: 6, c: 6

h: 8

Isosceles Triangle Base

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Calculate properties of a prism with an isosceles triangle base.

a: 5, b: 5, c: 8

h: 12

Scalene Triangle Base

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Calculate properties of a prism with a scalene (all sides different) triangle base.

a: 7, b: 9, c: 11

h: 15

Other Titles
Understanding the Triangular Prism Calculator: A Comprehensive Guide
Explore the geometry of triangular prisms, from volume and surface area calculations to real-world applications and mathematical derivations.

What is a Triangular Prism? Core Concepts

  • A triangular prism is a three-dimensional shape with two parallel triangular bases and three rectangular sides.
  • The orientation of the rectangular sides determines if it's a right prism or an oblique prism.
  • Its properties are defined by the dimensions of its base and its height.
A triangular prism is a member of the prism family of polyhedra. It is characterized by having two identical and parallel triangular faces, called the bases, which are connected by three rectangular faces, known as the lateral faces. The shape of the triangular base can be any type of triangle: equilateral, isosceles, scalene, or right-angled.
Types of Triangular Prisms
Right Triangular Prism: In a right prism, the lateral faces are rectangles and are perpendicular to the triangular bases. This is the most common type of triangular prism.
Oblique Triangular Prism: In an oblique prism, the lateral faces are parallelograms, and the prism appears 'slanted' because the bases are not directly aligned one over the other.
This calculator is designed for right triangular prisms, where the height 'h' is the perpendicular distance between the two bases.

Step-by-Step Guide to Using the Triangular Prism Calculator

  • Enter the three side lengths of the triangular base.
  • Input the height of the prism.
  • Instantly get the calculated volume, base area, lateral surface area, and total surface area.
Our calculator simplifies the process of finding the key properties of a triangular prism. Follow these steps for an accurate calculation.
Input Fields
1. Base Side 'a', 'b', and 'c': These fields represent the lengths of the three sides of the prism's triangular base. You must enter a positive number for each.
2. Prism Height 'h': This is the perpendicular distance between the two triangular bases. Enter a positive value for the height.
Interpreting the Results
  • Volume: The total space enclosed by the prism.
  • Base Area: The area of one of the triangular bases. The calculation uses Heron's formula based on the side lengths.
  • Lateral Surface Area: The combined area of the three rectangular side faces.
  • Total Surface Area: The sum of the areas of the two triangular bases and the three rectangular faces.

Calculation Examples

  • Input: a=3, b=4, c=5, h=10 -> Base Area = 6, Volume = 60
  • Input: a=6, b=6, c=6, h=8 -> Base Area ≈ 15.59, Volume ≈ 124.7

Mathematical Formulas and Derivations

  • The volume is the product of the base area and the prism's height.
  • The surface area is the sum of the areas of the two bases and the lateral faces.
  • Heron's formula is used to find the area of the triangular base from its side lengths.
Understanding the formulas behind the calculations can provide deeper insight into the geometry of a triangular prism.
1. Base Area (A_base)
When the side lengths (a, b, c) are known, the area of the triangle can be found using Heron's Formula. First, we calculate the semi-perimeter (s):
s = (a + b + c) / 2
Then, the area is:
A_base = sqrt(s * (s - a) * (s - b) * (s - c))
2. Volume (V)
The volume is the base area multiplied by the prism's height (h):
V = A_base * h
3. Lateral Surface Area (A_lateral)
The lateral surface area is the sum of the areas of the three rectangular faces. The area of each rectangle is its side length (from the triangle base) times the prism height. Therefore, the total lateral area is the perimeter of the base multiplied by the height.
Perimeter_base = a + b + c
A_lateral = Perimeter_base * h = (a + b + c) * h
4. Total Surface Area (A_total)
The total surface area is the sum of the lateral area and the area of the two triangular bases:
A_total = A_lateral + 2 * A_base

Real-World Applications of Triangular Prisms

  • Architecture and Construction: Used in roof structures and modern building designs.
  • Optics: Prisms are used to disperse light into its constituent colors.
  • Camping and Recreation: The classic shape of a camping tent is a triangular prism.
Architecture
The most common architectural application of triangular prisms is in the gabled roofs of houses. The shape provides stability and allows for effective water drainage. Modern architecture also utilizes this shape for aesthetic and structural purposes in atriums and facades.
Science and Optics
In physics, glass or crystal triangular prisms are fundamental tools in optics. When white light passes through a prism, it undergoes refraction, splitting it into the colors of the rainbow (a spectrum). This phenomenon, known as dispersion, was famously studied by Isaac Newton and is crucial for spectroscopy.
Everyday Objects
Many everyday objects take the shape of a triangular prism. A classic camping tent, a slice of cheese or cake, and some types of packaging (like a Toblerone chocolate bar box) are all examples of triangular prisms in our daily lives.

Common Questions and Key Considerations

  • A valid triangle must satisfy the triangle inequality theorem.
  • All input values must be positive numbers.
  • The calculator assumes a 'right' prism, where side faces are rectangular.
What is the Triangle Inequality Theorem?
For any three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. The calculator validates this condition (a+b > c, a+c > b, and b+c > a) to ensure the inputs are geometrically possible. If the condition is not met, it will display an error.
Can I use this for an oblique prism?
This calculator is specifically designed for right triangular prisms. For an oblique prism, the calculation of the lateral surface area is more complex as the sides are parallelograms, not rectangles. The volume calculation, however, remains the same as long as 'h' is the perpendicular height.
What if my base is a right-angled triangle?
The calculator works perfectly for right-angled triangles. Simply enter the three side lengths as you would for any other triangle. For example, for a triangle with sides 3, 4, and 5 (where 5 is the hypotenuse), you would enter these three values into the 'a', 'b', and 'c' fields.