The base of a prism has 𝑛 sides. Circle the expression for the number of faces of the prism. Is it two 𝑛, 𝑛 plus two, 𝑛 plus one, or two 𝑛 plus two?
Remember, a prism is a three-dimensional shape which has a constant cross section. The cross section is the view of the inside of the shape when it’s sliced through. A cone, for example, is not a prism. No matter which direction we slice, the cross section always changes. A cylinder is a prism though. If we slice it horizontally parallel to the circular face, we get a circle as the cross section. And that circle is the same size no matter where we slice this cylinder horizontally.
Let’s see if we can find a relationship between the number of sides the base of the prism has — that’s just another way of saying the cross section — and the number of faces of that prism. The base of our cylinder is a circle. It has one side. The prism though has three faces. It has the two circles, and it has the curved surface which forms a rectangle.
Next, let’s look at a triangular prism. The cross section or the base of a triangular prism is a triangle. And the a has three sides. A triangular prism though has five faces. It has two triangles at each end, and then there are three rectangles.
In fact, we can see in each of these cases that the number of faces is two more than the number of sides the base of the prism has. We can think about it like this. We now know that all prisms have to have two identical faces. Those are the faces at each end of the shape that are parallel to the cross section. Then there’s a face coming off of each side of this cross section. That means there is a face for each side plus two more faces. In the case of an 𝑛-sided shape, this is 𝑛 plus two.
We can check this by considering a cuboid. Another way of describing a cuboid is to call it a rectangular prism. It has a base which is a rectangle, and a rectangle has four sides. We know that a cuboid has six faces. So this cuboid satisfies our expression. The expression for the number of faces the prism has is 𝑛 plus two.