A Cube with a Whole-Number Edge

A wooden block is a cube with an edge of $5$ cm. What is its volume?

Write the cube formula: $V = s^3 = s \times s \times s$ = Formula ready

Substitute the edge: $V = 5 \times 5 \times 5$ = Value plugged in

Multiply the first two edges: $5 \times 5 = 25$ = Area of one face: $25$ cm²

Multiply by the third edge: $25 \times 5 = 125$ = $V = 125$ cm³

Answer: The volume of the block is $125$ cm³.

A Cube with a Decimal Edge

A cube-shaped garden planter has an edge of $2.5$ m. What is its volume?

Write the formula with the edge: $V = 2.5 \times 2.5 \times 2.5$ = Setup complete

Multiply the first two numbers: $2.5 \times 2.5 = 6.25$ = Area of one face: $6.25$ m²

Multiply by the third edge: $6.25 \times 2.5 = 15.625$ = $V = 15.625$ m³

Check with an estimate: $2^3 = 8$ and $3^3 = 27$, so the answer must be between $8$ and $27$ = $15.625$ makes sense

Answer: The planter holds $15.625$ m³ of soil.

Finding the Edge from the Volume

A cube has a volume of $343$ cm³. How long is its edge?

Write what you need: $s \times s \times s = 343$, so $s = \sqrt[3]{343}$ = Find a number that cubes to 343

Try a guess: $6^3 = 216$, too small = The edge is more than 6

Try the next whole number: $7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$ = It matches!

Answer: The edge of the cube is $7$ cm.

What Happens When the Edge Doubles?

Cube A has an edge of $3$ cm. Cube B has an edge of $6$ cm, twice as long. How many times larger is the volume of cube B?

Find the volume of cube A: $3^3 = 3 \times 3 \times 3 = 27$ = $V_A = 27$ cm³

Find the volume of cube B: $6^3 = 6 \times 6 \times 6 = 216$ = $V_B = 216$ cm³

Divide the volumes: $216 \div 27 = 8$ = B is 8 times larger

See why: $(2s)^3 = 2s \times 2s \times 2s = 8s^3$ = Each of the 3 edges doubles: $2 \times 2 \times 2 = 8$

Answer: Doubling the edge makes the volume $8$ times larger, not 2 times.

From cm³ to Liters

A cube-shaped water tank has an inside edge of $30$ cm. How many liters of water does it hold? ($1$ L $= 1000$ cm³)

Multiply the first two edges: $30 \times 30 = 900$ = Area of the bottom: $900$ cm²

Multiply by the height: $900 \times 30 = 27000$ = $V = 27000$ cm³

Convert to liters: $27000 \div 1000 = 27$ = $27$ L

Check with the 1 liter cube: $30$ cm is $3$ times $10$ cm, so the tank holds $3^3 = 27$ one-liter cubes = Same answer

Answer: The tank holds $27$ liters of water.

Mistake: Multiplying the edge by 3: for $s = 5$, writing $V = 5 \times 3 = 15$

Why: $s^3$ means $s$ multiplied by itself 3 times, not $s$ times 3.

Correct: $5^3 = 5 \times 5 \times 5 = 125$ cm³.

Mistake: Stopping after two edges: $5 \times 5 = 25$

Why: $s \times s$ is the area of ONE face (square units). Volume needs all three dimensions.

Correct: Multiply by the edge one more time: $25 \times 5 = 125$ cm³, and write cubic units.

Mistake: Thinking that doubling the edge doubles the volume

Why: The edge appears three times in $s \times s \times s$, so each of the three factors doubles.

Correct: Doubling the edge multiplies the volume by $2^3 = 8$. Tripling it multiplies the volume by $3^3 = 27$.

Mistake: Misplacing the decimal point: $0.2^3 = 0.8$ or $0.6$

Why: Each multiplication by a decimal adds decimal places: one decimal place three times gives three decimal places.

Correct: $0.2 \times 0.2 = 0.04$, then $0.04 \times 0.2 = 0.008$. A small edge gives an even smaller volume.

Ice Cube Trays

Ice cube trays make small cubes of ice. Knowing the volume of one cube tells you how much water a whole tray needs.

An ice cube with a $2$ cm edge holds $2^3 = 8$ cm³ of water.

Storage Cubes

Fabric storage cubes for shelves come in cube shapes. Their volume in liters tells you how much they can hold.

A storage cube with a $10$ cm edge holds $1000$ cm³, which is exactly $1$ liter.

Toy Building Blocks

Toy companies pack small cube blocks into larger cube boxes. The box must hold a whole number of blocks along each edge.

A box with a $6$ cm edge holds $3 \times 3 \times 3 = 27$ blocks with a $2$ cm edge.

Q: Why is $s^3$ called "s cubed"?

A: Because $s \times s \times s$ is exactly the volume of a cube with edge $s$. The name of the power comes from the shape!

Q: Is a cube a rectangular prism?

A: Yes. A cube is a rectangular prism where length, width and height are equal. That is why $V = l \times w \times h$ becomes $V = s \times s \times s$.

Q: How do I find the edge if I only know the volume?

A: Find the number that multiplied by itself 3 times gives the volume. That is the cube root: $\sqrt[3]{125} = 5$ because $5^3 = 125$. Learning the perfect cubes up to $1000$ makes this fast.

Q: What if the edge is a fraction?

A: The formula works the same way: $\left(\frac{1}{2}\right)^3 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$. A cube with a half-unit edge has one eighth of a cubic unit of volume.