A box with edges in halves can be packed with cubes measuring a half on each side.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Count the little cubes
A box 3 halves by 2 halves by 2 halves holds 3 × 2 × 2 = 12 small cubes.
What each little cube is worth
Eight of them fill a unit cube, so each is one eighth. 12 of them is 12/8, which is 1½.
The formula agrees
1½ × 1 × 1 = 1½. Counting cubes and multiplying edges give the same answer.
Counting the small cubes
Fill a box with cubes of side 1/2 and count them. Each small cube has volume 1/8 of a unit cube, since 8 of them fit in one. Counting the cubes and multiplying by 1/8 gives the same answer as the formula — which is the point of the exercise.
Two methods, one answer
A box 1 by 1 by 1/2 holds 4 half-cubes, so its volume is 4 × 1/8 = 1/2. The formula gives 1 × 1 × 1/2 = 1/2. When counting and formula agree, you have evidence the formula really does describe the packing.
The cube size must divide the edges
Half-cubes pack a box with edges that are multiples of a half. If an edge were 1/3, half-cubes would not fit without cutting. Packing arguments need a cube size that divides every edge, which is why the examples are chosen as they are.
Why do the counting at all
The formula would have given the answer directly. The counting is there to show the formula is not an arbitrary rule that happens to work for whole numbers — it describes what is physically happening, at any scale.
Step 2: Try It Yourself
Tap and try it out.
- Cubes in one layer3 × 2 = 6
- Layers2
- 3 × 2 × 212 cubic units
Counting every cube gives 12, and so does 3 × 2 × 2. The formula is a shortcut for the count.
Step 3: Watch an Example
One step at a time.
Watch Leo Pack a Box
A box is 1½ by 1 by 1, packed with half-sized cubes.
- Step 1
He counts along the length: 1½ is three halves, so 3 cubes.
Step 4: Your Turn
Practice makes it stick.
The Small Cubes
Problem 1 of 2
How many cubes with half-unit edges fill one whole unit cube?
The Packed Box
Problem 2 of 2
A box holds 24 half-unit cubes. What is its volume in cubic units?
Count and Convert
1 of 4
4 by 2 by 2 small cubes. How many cubes?
2 of 4
16 half-unit cubes. Volume in cubic units?
3 of 4
How many third-unit cubes fill one unit cube?
4 of 4
A box 2 by 3 by 4 units. Volume?
Step 5: Quick Check
Show what you know.
Question 1 of 1
32 half-unit cubes. Volume in cubic units?
What You Learned
- A box with fractional edges can be packed with unit-fraction cubes.
- Eight half-unit cubes fill one unit cube; twenty-seven third-unit cubes do.
- Counting cubes and multiplying edges give the same volume.