Lesson 6 of 8 · Grades 3 to 6
Volume by counting cubes
Volume measures the space inside a solid. A unit cube is 1 unit long, 1 unit wide and 1 unit tall, and the volume of a solid counts how many unit cubes fill it.
Unit cubes
Area measures a flat surface. Volume measures the space inside a solid, and it works the same way: cover the space with equal cubes and count them. A unit cube is 1 unit long, 1 unit wide and 1 unit tall. If the units are centimetres, each cube has a volume of 1 cm³.
The box below is built from unit cubes. It is 4 cubes long, 3 cubes wide and 2 cubes tall. Counting every cube one at a time would work, but there is a faster way. Look at the layers.
Count one layer, then multiply
The bottom layer of the box is a rectangle of cubes. It is 4 cubes long and 3 cubes wide, so it holds 4 × 3 = 12 cubes. The box is 2 layers tall, so there are 12 + 12 = 24 cubes, or simply 12 × 2 = 24.
The same plan works for any box. Find the number of cubes in one layer, then multiply by the number of layers. A box 5 cubes long by 2 cubes wide holds 10 cubes in each layer. If it is 3 layers tall, it holds 30 cubes in total.
Cubic units
The volume of the box counts 24 unit cubes, so its volume is 24 cubic units. A cubic centimetre, written cm³, is a cube 1 cm on every edge. A cubic metre, written m³, is a cube 1 m on every edge. The little 3 shows that three lengths were multiplied.
Cubic units are much bigger than the square units used for area. A box that is 10 cm long, 10 cm wide and 10 cm tall holds 1,000 cm³, because each layer holds 100 cubes and there are 10 layers.
Building bigger solids
Volume grows quickly when the edge lengths grow. Doubling every edge of a box makes the volume 8 times bigger, because each row doubles, the number of rows doubles, and the number of layers doubles too: 2 × 2 × 2 = 8.
When a solid is not a plain box, count what you can see. Split it into box-shaped parts, count the cubes in each part, then add. The last lesson of the course uses this plan on real objects.
Worked example
A box is built from unit cubes: 4 cubes long, 3 cubes wide and 3 cubes tall. How many cubes fill the box?
Step 1: count one layer of cubes. The layer is 4 long and 3 wide, so 4 × 3 = 12 cubes.
Step 2: count the layers. The box is 3 cubes tall, so there are 3 equal layers.
Step 3: add the layers by multiplying. 12 × 3 = 36.
Step 4: check with the other order. 4 × 3 × 3 also gives 36, so the box holds 36 unit cubes.
Answer: 36 unit cubes.
Counting tip
Always count a full layer before you count the layers. Write the layer number down: it stops you losing your place in a tall stack. If the solid has a part stuck on the side, count that part as its own little box and add it at the end.
Remember
- Volume counts the unit cubes that fill a solid.
- Find the cubes in one layer, then multiply by the layers.
- Cubic units, such as cm³ and m³, measure volume.
- Split an odd solid into boxes, count each box, then add.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
-
A box is 5 cubes long, 2 cubes wide and 1 cube tall. How many unit cubes fill it?
Show answer
10 unit cubes. 5 × 2 = 10, and one layer means 10 cubes.
-
A layer holds 8 cubes. The box is 3 layers tall. How many cubes fill it?
Show answer
24 cubes. 3 × 8 = 24.
-
A box is 4 cm by 3 cm by 2 cm built from 1 cm cubes. How many cubes fill it?
Show answer
24 cubes. 4 × 3 × 2 = 24.
-
A cube is 3 cm on every edge. How many 1 cm cubes fill it?
Show answer
27 cubes. 3 × 3 × 3 = 27.
-
Which holds more cubes: a box 5 cubes long by 2 wide by 2 tall, or a cube 3 cubes on every edge?
Show answer
The cube. The box holds 20 cubes, and the cube holds 27.
Puzzle · Cube count
Puzzle time
A box is built from unit cubes. The bottom layer is 4 cubes long and 3 cubes wide, and the box is 2 layers tall. How many unit cubes fill the box?
Show a hint
Count one layer first: 4 × 3. Then multiply by the number of layers.
Show the answer
Answer: 24 cubes. One layer holds 4 × 3 = 12 cubes. There are 2 layers, and 12 × 2 = 24, so the box holds 24 unit cubes.