How to Find the Volume and Surface Area of a Prism

What this lesson covers

A prism is a solid whose two ends are the same shape, like the same slice all the way through. A box is a rectangular prism. A triangular prism has a triangle at each end and three rectangles between them.

This lesson teaches volume, the space inside a solid, and surface area, the total area of all its faces. It also covers nets, the flat figures you get when you unfold a solid.

On the ISEE Middle Level math sections, these come as multiple-choice questions and as quantitative comparisons, with no calculator.

The method

Volume = base area × height.

1. The base is one of the two matching ends. Its area counts the 1-unit cubes in one layer.
2. The height is the distance between the two ends. It counts the layers.
3. Multiply. For a box, that's length × width × height, in cubic units.

For a triangular prism, the base is a triangle end, even when the prism lies on its side, and the height is the length between the two triangles. A triangle's area is its base times its height, divided by 2.

Surface area from the net.

1. Unfold the solid into its net.
2. Match each face of the net to a face of the solid.
3. Find the area of each face, and add them all, in square units.

A box has 6 faces in three matching pairs: the top and the bottom, the front and the back, and the two sides. A cube has 6 equal square faces.

Worked example

A box is 6 centimeters long, 4 centimeters wide and 5 centimeters tall. How many 1-centimeter cubes fill it, and what is its surface area?

Start with volume. The bottom layer is 6 cubes long and 4 cubes wide, so it holds $6 \times 4 = 24$ cubes. The box is 5 centimeters tall, so there are 5 layers: $24 \times 5 = 120$. It holds 120 cubes, and its volume is 120 cubic centimeters.

Now surface area. Pair up the six rectangles of its net. The top and the bottom are each 6 by 4, which is 24 square centimeters. The front and the back are each 6 by 5, which is 30. The two sides are each 4 by 5, which is 20. One face of each pair adds to $24 + 30 + 20 = 74$. Both faces of each pair make $2 \times 74 = 148$. The surface area is 148 square centimeters.

Volume counts cubes inside, so it is in cubic centimeters. Surface area counts squares on the outside, so it is in square centimeters.

The common mistake

The lesson's common mistake is leaving out the hidden faces. A picture of a box shows only three faces at once.

Say a box is 4 inches long, 2 inches wide and 2 inches tall. A student adds the three faces in the picture: the top is 4 by 2, the front is 4 by 2, and the side is 2 by 2. That gives $8 + 8 + 4 = 20$ square inches.

But a box has 6 faces. The bottom, the back and the other side are hidden. Each hidden face matches the face opposite it, so the three hidden faces also add to 20. The surface area is $2 \times 20 = 40$ square inches.

To avoid this, count all six faces, or unfold the box into its net so every face is in view.