Prisms and Their Views

A prism is a solid with the same cross-section all the way through. Imagine slicing it like a loaf of bread: if every slice gives you exactly the same flat shape, you have a prism, and that repeating shape is called the cross-section. A box gives the same rectangle on every slice, so it is a rectangular prism. A wedge of cheese gives the same triangle, so it is a triangular prism. A pyramid and a cone fail the test, because their slices shrink towards a point as you move up. One thing to be careful about: the cross-section is not always the face sitting on the ground. A triangular prism resting on a flat side still has the triangle as its cross-section.

To describe a solid on paper you draw it from three directions: the front, the side and the top. A view is what you see looking straight at the solid from that direction, and it is always flat — no corners running off at an angle, no shading, just the outline you would see. Here is the useful part. This box is 8 centimetres wide, 3 deep and 5 high, and each view shows only two of those three sizes. From the front you see the width and the height. From the side you see the depth and the height. From above you see the width and the depth. The third size is hidden behind what you are looking at.

Now draw the three views of a real box, 8 centimetres wide, 3 deep and 5 high. Every view of a box is a rectangle, and you build each one by taking the sizes two at a time. The front view uses the width and the height, so it is 8 by 5. The side view uses the depth and the height, so it is 3 by 5 — much narrower. The top view uses the width and the depth, so it is 8 by 3 — wide but shallow. Draw all three to the same scale and set them out side by side. Notice the front and top views are both 8 centimetres across, so their edges line up, and that lining up is your check.

A triangular prism is more interesting, because only one of its views is a triangle. This one has a triangle 6 centimetres across and 4 centimetres tall, and the prism runs 10 centimetres back. Looking from the front you see the triangle itself, 6 by 4. Looking from the side you see a flat rectangle 10 long and 4 high — the triangle has vanished, because from there you are looking along it. Looking from above you see a rectangle 6 across and 10 back, with a line down the middle where the top ridge sits. Keep the two lengths apart in your head: the triangle is 4 tall, the whole prism is 10 long.

Here is the check that catches mistakes. Every measurement of a box appears in exactly two of the three views. The width of 8 shows in the front and the top, the height of 5 in the front and the side, and the depth of 3 in the side and the top. If a number turns up in only one of your views, one of them is drawn wrong. The rule also works backwards. Suppose the front view is 10 by 6 and the side view is 4 by 6. The 6 appears in both, so 6 is the height; the front gives a width of 10 and the side gives a depth of 4. So the solid is a box 10 by 4 by 6.