Volume: build it one layer at a time

A box takes up space

Area tells us how much flat surface is covered. Volume tells us how much three-dimensional space is occupied. For our rectangular prisms, we count unit cubes. Each small cube measures one centimeter along every edge, so its volume is one cubic centimeter.

Build the bottom layer

Start with a base four centimeters long and two centimeters wide. One layer has two rows of four unit cubes, making eight cubes. There are no gaps or overlaps in this model. We are counting all the cubes in a layer, not only the ones around its edge.

Stack equal layers

Stack three of those layers. Eight cubes in each layer, times three layers, makes twenty-four cubes. The diagram spreads the layers apart so you can count them. Imagine stacking them into one prism, not leaving them as three separate mats on the floor.

Keep the dimensions straight

For this rectangular prism, length times width counts cubes in a layer. Multiplying by height counts all the layers. Four times two times three is twenty-four cubic centimeters. Adding four, two, and three would answer a different question. A box is not filled by adding edge labels.

Pause: find a missing height

Try a new prism. Its base is five centimeters by four centimeters, and its volume is sixty cubic centimeters. Pause and find the height. First count the unit cubes in one one-centimeter-high layer. Then work out how many such layers are needed.

Divide by the size of one layer

Five times four gives twenty cubes in each one-centimeter-high layer. Sixty divided by twenty is three layers, so the height is three centimeters. Check by multiplying five times four times three. You get sixty cubic centimeters again, matching the given volume.

Turn the prism; keep the volume

Fun fact: turning this prism onto another face changes which dimension you call height, but not the amount of space it occupies. Four times two times three and three times four times two both give twenty-four. The cubes do not multiply just because the box takes a nap on its side.

Build, count, and explain

Continue to the volume worksheet. Sketch a layer, count how many layers there are, and label your answer in cubic units. For a missing height, divide volume by base area. For separate non-overlapping blocks, find each volume before adding their totals.