Chapter 11 · Volume
Volume: Counting Unit Cubes Layer by Layer
Learn what volume means and how it differs from length and area. Then find the volume of a solid by counting its unit cubes layer by layer, hidden ones included.
You already measure length in units and area in square units. Now you will learn a third kind of measure.
A solid figure is a shape that is not flat, such as a box, a ball or a stack of blocks. It takes up space. Volume is the amount of space a solid figure fills. A flat shape, such as a square drawn on paper, has side lengths and an area. But it fills no space, so it has no volume.
To measure volume, you need a unit that fills space. A cube is a solid figure with 6 flat, square sides, called faces, that are all the same size. An edge is a line segment where two faces meet. A unit cube is a cube whose edges are each 1 unit long.
One unit cube fills 1 cubic unit of space. Cubic units are the units for volume, just as square units are the units for area. You can use any unit of length. For example, a cube with edges 1 centimeter long fills 1 cubic centimeter. The table below shows how one unit cube has all three measures.
| Measure | What it tells you | Measured in | In one unit cube |
|---|---|---|---|
Length | How long something is, such as an edge | units | Each edge is 1 unit long. |
Area | How much flat surface a shape covers | square units | Each face covers 1 square unit. |
Volume | How much space a solid figure fills | cubic units | The whole cube fills 1 cubic unit. |
Key concept
Count unit cubes, layer by layer, to find volume
Picture unit cubes packed into a solid figure so that they fill it exactly. There are no gaps: no empty space is left between the cubes. There are no overlaps: no two cubes share the same space. When both are true, the number of cubes is the volume in cubic units. So a solid filled this way by 6 unit cubes has a volume of 6 cubic units.
To count the cubes, go one layer at a time. A layer is one flat level of cubes, all at the same height. Layers help you keep track of cubes that hide under or behind other cubes. Count every cube in each layer, even the ones you cannot see. Then add the layer totals. Each cube is in just one layer, so you count each cube once.
A solid made of unit cubes, 2 columns across and 2 rows deep. Layer 1: column 1, row 1; column 2, row 1; column 1, row 2; column 2, row 2. Layer 2: column 1, row 1; column 2, row 1.
Worked example
Count two layers
The solid shown above is built from unit cubes. Find its volume.
Start with the bottom layer, the cubes that rest on the ground. It has 2 rows with 2 cubes in each row. Add the rows to count the cubes in this layer.
Next, count the top layer. It has 2 cubes. Each one sits right on a bottom cube and covers its top. Those covered bottom cubes still fill space, and you already counted them in the bottom layer.
Add the two layer totals. Each cube is in just one layer, so you count every cube once. These 6 cubes fill the solid with no gaps and no overlaps.
Each unit cube fills 1 cubic unit, so 6 unit cubes fill 6 cubic units. You counted cubes, not squares, so the answer is in cubic units, not square units.
Answer
The volume of the solid is cubic units.
A solid made of unit cubes, 3 columns across and 3 rows deep. Layer 1: column 1, row 1; column 2, row 1; column 3, row 1; column 1, row 2; column 2, row 2; column 3, row 2; column 1, row 3; column 2, row 3; column 3, row 3. Layer 2: column 1, row 1; column 2, row 1; column 1, row 2; column 2, row 2. Layer 3: column 1, row 1.
Worked example
Three layers and a hidden cube
The solid shown above is built from unit cubes in three layers. Find its volume.
Count the bottom layer: 3 rows with 3 cubes in each row. Look at its center cube. It has a cube on each of its four sides, a cube on top and the ground below. In the solid, you cannot see it from any direction. It still fills space, so you count it.
Count the second layer. It sits on one corner of the bottom layer and has 2 rows with 2 cubes in each row.
The top layer has 1 cube. Add the three layer totals. The sum is the number of unit cubes, which is the volume in cubic units.
Answer
The volume of the solid is cubic units.
A solid made of unit cubes, 3 columns across and 3 rows deep. Layer 1: column 1, row 1; column 2, row 1; column 3, row 1; column 1, row 2; column 2, row 2; column 3, row 2; column 1, row 3; column 2, row 3. Layer 2: column 2, row 1; column 1, row 2; column 2, row 2; column 3, row 2; column 2, row 3. Layer 3: column 2, row 1; column 2, row 2.
Worked example
An uneven solid
The solid shown above is built from unit cubes. Its layers have different shapes. Find its volume.
Bottom layer: 2 rows of 3 cubes and 1 row of 2 cubes. Count the center cube even though it is hidden, because it fills space.
Second layer: rows of 1, 3 and 1 cubes, in a plus shape. The center cube of the plus is hidden too. It has cubes on all four sides, one above it and one below it. Count it.
Top layer: 2 cubes. Add the layer totals, since each cube is in just one layer.
Answer
The volume of the solid is cubic units.
Takeaway: Volume is the amount of space a solid figure fills. Length uses units and area uses square units, but volume uses cubic units, and one unit cube fills 1 cubic unit. To find the volume of a solid built from unit cubes, count the cubes one layer at a time, include the hidden ones, and add the layer totals.