Triangular prism volume calculator

What is a triangular prism?

A triangular prism is a three-dimensional solid object with two identical triangular bases and three rectangular lateral faces. It is an example of a prism where the cross-section perpendicular to the length is a triangle. Triangular prisms are frequently encountered in geometry and have applications in various fields such as architecture, art, and engineering. When you want to find the volume of a triangular prism, you’re essentially calculating how much space it occupies.

Types of triangular prisms

1. Regular triangular prism: Both triangular bases are equilateral.
2. Irregular triangular prism: The bases can be any triangle, including scalene or isosceles.
3. Rectangular triangular prism: Often refers to prisms with right-angled triangular bases.

Calculating the volume

The volume of a triangular prism can be calculated using different parameters as specified below. The fundamental formula for the volume of a triangular prism is:

where $V$ is the volume, $A_{\text{base}}$ is the area of the triangular base, and $L$ is the length of the prism.

1. Using the length of the prism and three sides of the triangle

For a triangle with sides $a$, $b$, and $c$, the area $A_{\text{base}}$ can be determined using Heron’s formula:

Thus, the volume becomes:

2. Using the length of the prism, two sides, and the included angle

For a triangle with sides $a$ and $b$, and the included angle $\theta$, the area $A_{\text{base}}$ is:

So the volume is:

3. Using the length of the prism, two angles, and the included side

Given a side $a$, and angles $\alpha$ and $\beta$, the third angle $\gamma$ can be found using:

The area using the Law of Sines is:

The volume becomes:

4. Using the length of the prism, base, and height

For a triangle with known base $b$ and height $h$:

Therefore, the volume is:

Examples

Example 1: Regular triangular prism

A regular triangular prism with a triangular base of sides 6 cm, 6 cm, and 6 cm, and a length of 10 cm.

• Calculate semi-perimeter: $s = \frac{6 + 6 + 6}{2} = 9 \text{ cm}$
• Using Heron’s formula: $$A_{\text{base}} = \sqrt{9(9-6)(9-6)(9-6)}$$ $$A_{\text{base}} = \sqrt{9 \times 3 \times 3 \times 3} = 9 \sqrt{3} \text{ cm}^2$$
• Volume: $$V = 9 \sqrt{3} \times 10 = 155.9 \text{ cm}^3$$

Example 2: Irregular triangular prism

For a triangular base with sides 8 cm, 5 cm, and 7 cm, and prism length of 12 cm.

• $s = \frac{8 + 5 + 7}{2} = 10 \text{ cm}$
• Heron’s formula: $$A_{\text{base}} = \sqrt{10(10-8)(10-5)(10-7)} = \sqrt{10 \times 2 \times 5 \times 3} \approx 17.32 \text{ cm}^2$$
• Volume: $$V = 17.32 \times 12 = 207.85 \text{ cm}^3$$

Example 3: Rectangular triangular prism

A triangular base with base 5 cm and height 6 cm, and the prism’s length is 15 cm.

• $A_{\text{base}} = \frac{1}{2} \times 5 \times 6 = 15 \text{ cm}^2$
• Volume: $$V = 15 \times 15 = 225 \text{ cm}^3$$

Notes

• Ensure all measurements are in the same unit before calculating.
• When calculating trigonometric functions, ensure the angle is in the correct unit (degrees or radians) as required.
• When using Heron’s formula, be careful with floating-point calculations to avoid precision errors.

Frequently Asked Questions

How to calculate the volume of a triangular prism with known side lengths?

To calculate the volume when the three sides of the triangle are known, use Heron’s formula to find the area of the triangular base and multiply by the prism’s length.

How many faces does a triangular prism have?

A triangular prism has five faces: two triangular bases and three rectangular lateral faces.

What is the difference between a regular and irregular triangular prism?

A regular triangular prism has bases that are equilateral triangles, whereas an irregular triangular prism can have bases with any triangular shape.

Can the length of the prism be shorter than the triangle’s longest side?

Yes, the length of the prism (often corresponding to the height in different orientations) can be shorter, longer, or even equal to any of the triangular base’s sides.