Measuring Volume: Counting Unit Cubes

Lesson 2 of 3: Measurement and Data

In this lesson:

• Count all cubes in a figure, including hidden ones
• Use cubic centimeters, cubic inches, and cubic feet
• Measure with improvised units and always name the unit

What You Will Learn About Volume

By the end of this lesson, you will:

1. Count all cubes, including hidden ones
2. Use cm³, in³, and ft³ correctly
3. Measure with improvised cube units
4. Apply layer counting for efficiency
5. Compare volumes using the same unit
6. Explain why different units give different numbers

How Many Cubes Are Hiding?

You already know that volume measures the space inside a solid figure.

• Volume = number of unit cubes that pack the figure
• Every cube counts — no gaps, no overlaps

But what happens when cubes stack up and some disappear from view?

Flat Figure: Every Cube Visible

A flat 4 × 3 layer has 12 unit cubes — all visible, easy to count.

Stacked Prism: Finding Hidden Cubes

• A 4 × 3 × 2 prism has two layers
• The top layer hides some bottom cubes
• Dotted lines reveal the hidden cubes underneath

The Layer Strategy: Counting Cubes Fast

Step 1: Count cubes in one layer

Step 2: Multiply by the number of layers

Layer counting finds hidden cubes without seeing them!

Your Turn: Count by Layers

A rectangular prism is 4 cubes long, 3 cubes wide, and 2 cubes tall.

• How many cubes in one layer?
• How many layers?
• What is the total volume?

Count the layer first, then multiply. Check the next slide for the answer.

Quick Check: How Many in This Prism?

How many cubes in a 5 × 3 × 2 figure?

• One layer: 5 × 3 = ?
• Two layers: ? × 2 = ?

Think about it before the next slide...

Counting Cubes in L-Shaped Figures

Not every figure is a perfect rectangular prism.

• An L-shape is two rectangles joined together
• Count each rectangular section separately
• Add the sections: total volume = section A + section B

Layer counting still works — just count each section's layers carefully.

Practice: Count All the Cubes

Find the volume of each figure using layers:

1. A 2 × 3 × 2 rectangular prism
2. A 4 × 2 × 3 rectangular prism
3. An L-shape: 3 × 2 × 2 block joined to 1 × 2 × 2 block

Write your answer in cubic units.

Answers: Counting All the Cubes

1. 2 × 3 × 2 = 12 cubic units
2. 4 × 2 × 3 = 24 cubic units
3. (3 × 2 × 2) + (1 × 2 × 2) = 12 + 4 = 16 cubic units

From Cubic Units to Real-World Units

So far we have counted "cubic units" — but what size is one cubic unit?

To measure in the real world, we need standard-sized cubes.

Next: the three standard cubic units you need to know.

Meet the Three Standard Cubic Units

Each unit cube is named for its edge length: cm³, in³, ft³.

Same Box, Different Units, Different Numbers

• Fill a box with cm cubes → count 24 cm³
• Fill the same box with in cubes → count 3 in³

The box did not change — the unit changed!

Smaller units → bigger numbers. Larger units → smaller numbers.

Matching Units to Real-World Objects

Rule: Pick the unit that gives a practical number.

Which Unit Would You Choose?

Match each object to the best unit: cm³, in³, or ft³

• A pencil case → ?
• A refrigerator → ?
• A sugar cube → ?

Think about the size of each object and each unit cube.

Practice: Count with Standard Units

1. A figure made of cubes with 1 cm edges: 3 × 4 × 2.
   Volume = ? cm³

2. A figure made of cubes with 1 in edges: 2 × 5 × 2.
   Volume = ? in³

Count by layers and include the correct unit.

Answers: Volume with the Correct Unit

1. 3 × 4 = 12 per layer, × 2 layers = 24 cm³
2. 2 × 5 = 10 per layer, × 2 layers = 20 in³

Always write the full unit: "24 cm³" not just "24."

Improvised Units: Any Cube Works

What if you don't have centimeter or inch cubes?

• Any cube-shaped object can be a unit cube
• Sugar cubes, connecting cubes, wooden blocks all work
• Two rules: the unit must be cube-shaped and consistent

Same Container, Two Different Units

• Small cubes: volume = 48 small cubes
• Large cubes: volume = 6 large cubes

Both measurements are correct! Same container, different units.

The number changes because the unit size changes.

Why Not Marbles? Units Must Be Cubes

• Marbles are spheres — they leave gaps when packed
• Gaps mean you are not measuring all the space
• Cubes pack perfectly — flat faces line up with no gaps

Only cube-shaped objects give an accurate volume count.

Quick Check: What Is Missing?

Sam says: "The volume of my pencil box is 20."

What is wrong with Sam's statement?

Think about what is missing before moving on...

Three Measurements of the Same Object

The same box was measured three times:

• 18 cubic centimeters (cm cubes)
• 3 cubic inches (in cubes)
• 45 connecting cubes (small cubes)

All three are correct. The object's size did not change — the unit changed each time.

Practice: Mixed Problems About Volume Measurement

1. A 3 × 3 × 3 cube: how many cubes total?
2. Leo counts 40 sugar cubes. Maria counts 5 large cubes. Same box. Who is correct?
3. Box A = 30 cm³. Box B = 25 in³. Which is larger?

Answers: Hidden Cubes and Unit Labels

1. 3 × 3 × 3 = 27 cubic units (not 19 — hidden cubes!)
2. Both are correct — same box, different units
3. Cannot compare directly — cm³ and in³ are different-sized units

Key Takeaways for Measuring Volume

✓ Count every cube — visible or hidden
✓ Layer strategy: cubes per layer × layers
✓ Write the full unit: cm³, in³, or ft³

Watch out: Hidden cubes still count
Watch out: Write "cm³" not "cm" — cubic means volume

What Comes Next in Volume Study?

Next lesson: Volume formulas — turning layer counting into

You already know the foundation:

• Cubes per layer = length × width
• Total volume = cubes per layer × height

The formula is just layer counting written as multiplication!