What is this skill?
Volume measures how much space a solid takes up. In fifth grade, children find the volume of rectangular boxes (called rectangular prisms) by thinking of them as built from unit cubes. A box that is 4 cubes long, 3 cubes wide and 5 cubes tall holds 4 × 3 = 12 cubes in each layer, and with 5 layers the total is 12 × 5 = 60 cubic units.
This gives the formula V = length × width × height. It can also be written as V = base area × height, where the base area is the length times the width. The practice asks for the volume from three measurements, and in the harder questions it asks for a missing height when the volume and the base are given. For example, if a box has a volume of 60 cubic centimeters and a base of 4 cm by 3 cm, the height is 60 ÷ 12 = 5 cm. Volume is always measured in cubic units, such as cubic centimeters, cubic inches or cubic feet.
Why it matters
Volume is the first time children measure in three dimensions, and it connects several earlier ideas: area from third grade, multiplication, and the meaning of a unit of measure. It prepares the ground for working with capacity in science, for packing and shipping problems, and for the surface area and volume of more complex solids in middle school.
It is also a fine place to see why a formula exists. Children who have built a box from blocks and counted the layers understand where length times width times height comes from, and will not need to memorize it.
Common mistakes
- Confusing volume with area or perimeter, such as adding the three numbers or multiplying only two of them.
- Leaving out the unit, or writing square units, not cubic ones. Area is in square units and volume is in cubic units.
- Having trouble seeing the layers inside a box in a flat drawing. Building a small one makes it concrete.
- Using the wrong operation for a missing side. Finding a missing height means dividing the volume by the base area.
- Mixing up units, for instance a length in inches and another in feet. All three have to be in the same unit before multiplying.
How to help at home
- Build boxes from blocks, sugar cubes or Lego bricks and count the cubes layer by layer.
- Measure a cereal box or a shoebox and work out its volume. Then fill it with small cubes or dried beans to compare.
- Ask “how many layers?” and “how many in each layer?” as the two parts of every problem.
- Compare two different boxes with the same volume, like 2 × 6 × 2 and 3 × 4 × 2. It shows that shapes can differ and still hold the same amount.
- Say the unit out loud: “cubic centimeters.” It reminds your child what is being counted.
Worked examples
These are real questions from the Tutorsaurus volume of boxes practice, with the answers worked out.
Example 1
A box is 3 cm long, 3 cm wide and 2 cm tall. What is its volume?
Answer: 18 cu cm. Volume = length × width × height = 3 × 3 × 2 = 18 cubic cm.
Example 2
A box is 8 m long, 7 m wide and 8 m tall. What is its volume?
Answer: 448 cu m. Volume = length × width × height = 8 × 7 × 8 = 448 cubic m.
Example 3
A box is 6 cm long, 6 cm wide and 2 cm tall. What is its volume?
Answer: 72 cu cm. Volume = length × width × height = 6 × 6 × 2 = 72 cubic cm.