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How to Find the Volume of a Triangular Prism

A common task in geometry is to find the volume of a triangular prism. This is often extremely tricky for students to understand, but the confusion is almost always due to one particular misunderstanding.

This article explains what that misunderstanding is, and shows the steps for solving problems on this topic. The article assumes that the reader has the appropriate prerequisite knowledge.

Instructions

    • 1

      The first step is to understand what a triangular prism is. Imagine that we have a two-dimensional triangle drawn on a piece of paper. The triangle can be any type (isosceles, right, equilateral, scalene, etc.). Now we'll extend the triangle "upwards" from the table. We form a three-dimensional solid which is known as a triangular prism. Try to visualize that the two end faces of it are triangles, and the three side faces are rectangles. As we extend it upwards, we are giving it what we could call height.

      We could also think of a triangle drawn on a chalkboard, and forming the triangular prism by extending it "outwards" from the board. We could think of that as giving it depth. In any case, we're dealing with a 2D triangle extended perpendicular to itself into the third dimension. Study the diagrams in this article to help you visualize it.

    • 2

      We now must understand what volume means. Just like area is the space inside a two-dimensional figure such as the triangle that we started with, volume is the space inside a three-dimensional solid such as a triangular prism. To compute volume we always must multiply three dimensions in some way.

    • 3

      Before we can find the volume of a triangular prism, we must first find the area of the triangle from which it was formed. The formula for the area of a triangle is half the base times height. See the Resource section for an entire article on just that topic. When we say height in that context, we mean a line drawn from the highest point of the triangle straight down and perpendicular to the base. The height is never measured along a slanted side of the triangle. See the diagrams in this article.

    • 4

      Once we know the area of the starting triangle, all we need to do is multiply that area times the third dimension. Here is where it gets a bit confusing. Many books label the third dimension as height, using a letter h just like we used for the height of the triangle. It doesn't actually matter what we call the third dimension. All we have to do is multiply that third dimension times the area of the triangle. It's important to understand that you may see two h's in diagrams for problems like this, but one has nothing to do with the other. In the diagrams of this article, one of the h's has been written in red to distinguish it. The height of the prism has nothing at all to do with the height of the starting triangle.

      Note that you may also see the third dimension represented as d for depth. That is certainly less confusing since it doesn't reuse the letter h for a different purpose, but most books seem to purposely reuse the letter h to make sure that students fully understand how to do problems like this.

    • 5

      With all that said, to find the volume of a triangle prism, we just compute the area of the starting triangle using the formula given above, and then multiply it by the third dimension, whether we call it d or a second use of the letter h. Remember to express your answer in cubic units which is what we always do with volume problems regardless of the shape of the solid involved. Keep practicing. ☺

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