Calculate The Volume of a Triangular Prism

🔑 Key Concepts

• The volume of any prism can be found with \(V=BH\), where \(B\) is the area of the base and \(H\) is the height or length of the prism.
• For a triangular prism, the base is a triangle, so \(B=\frac{1}{2}bh\).
• The complete formula can be written as \(V=\left(\frac{1}{2}bh\right)H\).
• A triangular prism can be viewed as half of a matching rectangular prism divided along a diagonal plane.
• Only the perpendicular base and height of the triangular face are used to calculate its area.
• Volume is measured in cubic units because it represents three-dimensional space.

💡 Main Idea

The volume of a triangular prism is found by multiplying the area of its triangular base by the height of the prism: \(V=BH\). Since the base is a triangle, its area is \(B=\frac{1}{2}bh\), so the formula can also be written as \(V=\left(\frac{1}{2}bh\right)H\). A triangular prism can also be understood as half of a matching rectangular prism. Dividing a rectangular prism along a diagonal plane creates two congruent triangular prisms, which explains the factor of \(\frac{1}{2}\).

✏️ Worked Example — Volume of a Triangular Prism

Find the Volume of the Triangular Prism

The triangular base has a base of \(6\text{ ft}\) and a perpendicular height of \(8\text{ ft}\). The prism extends \(7\text{ ft}\). Find its volume.

Important: The \(10\text{ ft}\) slanted side is not needed. Triangle area requires the perpendicular base and height, \(6\text{ ft}\) and \(8\text{ ft}\).

Step 1 — Find the Area of the Triangular Base

Use the triangle area formula:

\[ B=\frac{1}{2}bh \]

Substitute \(6\) for the triangle’s base and \(8\) for its perpendicular height.

\[ B=\frac{1}{2}(6)(8) \]

\[ B=24\text{ ft}^2 \]

Step 2 — Multiply by the Height of the Prism

The prism extends \(7\text{ ft}\), so substitute \(B=24\) and \(H=7\) into \(V=BH\).

\[ V=BH \]

\[ V=24(7) \]

\[ V=168 \]

\[ \boxed{V=168\text{ ft}^3} \]

The volume of the triangular prism is \(168\text{ cubic feet}\). Cubic units are used because volume measures the amount of three-dimensional space inside the prism.

📚 What You Should Already Know

Before working on this lesson, students should know how to find the area of a triangle using \(A=\frac{1}{2}bh\), identify perpendicular base and height measurements, and find the volume of a rectangular prism. Students should also understand that the height of a prism is the distance between its two congruent bases.

🚀 What Comes Next

Students can extend the formula \(V=BH\) to other prisms and cylinders, compare volume formulas across different solids, reason about cross sections, and connect the areas of two-dimensional bases to the volumes of three-dimensional figures.

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