Volume of Rectangular Prisms

📦 Volume of Rectangular Prisms

Part 1 of 5 — What Volume Means

Topics in This Part

🔑 Key Concept: Volume measures how much space a solid fills — how many unit cubes fit inside it. Every formula in this lesson is just a fast way to count those cubes.

What Is a Rectangular Prism?

A rectangular prism is a box-shaped solid. Every face is a rectangle, and it has three measurements:

• Length ($l$) — how long it is
• Width ($w$) — how deep it is
• Height ($h$) — how tall it is

Think of a cereal box, a brick, a shipping container, or a fish tank — all are rectangular prisms.

💡 A cube is a special rectangular prism where length, width, and height are all equal.

Concept Check 🎯

Counting Unit Cubes

A unit cube is a cube measuring $1 \times 1 \times 1$. Its volume is exactly one cubic unit.

To find a prism's volume by hand, you can imagine filling it with unit cubes and counting them.

Picture a box that is $4$ long, $3$ wide, and $2$ tall:

• The bottom layer is a $4 \times 3$ rectangle of cubes $= 12$ cubes.
• There are $2$ layers stacked on top of each other.
• Total $= 12 \times 2 = 24$ unit cubes.

$\text{Volume} = 24 \text{ cubic units}$

🔑 The shortcut: Counting one layer is just $l \times w$, and stacking $h$ layers multiplies by $h$. That gives the formula $V = l \times w \times h$ — coming in Part 2.

Count the Cubes 🧮

Each box is built from $1 \times 1 \times 1$ unit cubes. Find the volume in cubic units.

1) Bottom layer of $5 \times 2 = 10$ cubes, stacked $3$ layers high. Volume $= \,?$ 2) Bottom layer of $4 \times 4 = 16$ cubes, stacked $2$ layers high. Volume $= \,?$ 3) Bottom layer of $6 \times 1 = 6$ cubes, stacked $1$ layer high. Volume $= \,?$

Stack the Layers 🔽

A box has a bottom layer of $4 \times 5 = 20$ cubes and is stacked $3$ layers high. Choose each piece.

Why Volume Is "Cubic"

When we multiply three lengths together, we multiply their units too:

$\text{cm} \times \text{cm} \times \text{cm} = \text{cm}^3 \quad(\text{read "cubic centimeters"})$

That little $3$ exponent is your signal that the answer is a volume — not a length or an area.

⚠️ Watch your units. A volume answer always ends in cubic units like $\text{cm}^3$, $\text{in}^3$, or $\text{m}^3$. Leaving off the $3$ — or the unit entirely — is the most common way to lose a point.

📦 Volume of Rectangular Prisms

Part 2 of 5 — The Formula $V = lwh$

🔑 The Idea: Instead of counting cubes one by one, just multiply the three measurements. $V = l \times w \times h$ counts every cube at once.

The Volume Formula

$V = l \times w \times h$

where $l$ = length, $w$ = width, and $h$ = height. Because multiplication can be done in any order, it does not matter which side you call length, width, or height — the product is the same.

Worked Example: $l = 5$, $w = 3$, $h = 4$ (cm)

$V = 5 \times 3 \times 4 = 60 \text{ cm}^3$

A clean way to do it is two numbers at a time:

$V = (5 \times 3) \times 4 = 15 \times 4 = 60 \text{ cm}^3$

💡 Tip: Multiply two numbers first, then multiply that result by the third. It keeps the arithmetic simple.

Worked Example: A Fish Tank

A fish tank is $40\text{ cm}$ long, $20\text{ cm}$ wide, and $25\text{ cm}$ tall.

$V = 40 \times 20 \times 25$

Multiply the first two: $40 \times 20 = 800$. Then:

$V = 800 \times 25 = 20{,}000 \text{ cm}^3$

✅ Check: Since $8 \times 25 = 200$, we get $800 \times 25 = 20{,}000$. ✓

So the tank holds $20{,}000$ cubic centimeters of water.

Concept Check 🎯

Plug In and Multiply 🧮

Use $V = l \times w \times h$. Enter the volume as a number (units are given).

1) $l = 4$, $w = 3$, $h = 2$ (cm). $V = \,?\ \text{cm}^3$ 2) $l = 10$, $w = 5$, $h = 6$ (in). $V = \,?\ \text{in}^3$ 3) $l = 7$, $w = 7$, $h = 2$ (ft). $V = \,?\ \text{ft}^3$

Build the Calculation 🔽

A storage bin is $8\text{ in}$ long, $4\text{ in}$ wide, and $3\text{ in}$ tall. Choose each piece.

📦 Volume of Rectangular Prisms

Part 3 of 5 — Base Area × Height

🔑 A second view: $V = lwh$ can be grouped as $V = (l \times w) \times h$. The part $l \times w$ is the area of the base, so $V = B \times h$ where $B$ is the base area.

Volume = Base Area × Height

$V = B \times h \qquad\text{where } B = l \times w$

This is exactly the "layers" idea from Part 1: the base area $B$ is the number of unit squares in one layer, and $h$ is the number of layers.

Worked Example: $l = 6$, $w = 4$, $h = 5$ (cm)

Step 1 — Base area: $B = l \times w = 6 \times 4 = 24 \text{ cm}^2$

Step 2 — Multiply by height: $V = B \times h = 24 \times 5 = 120 \text{ cm}^3$

💡 Notice the base area is in square units ($\text{cm}^2$), and multiplying by the height adds the third dimension to make cubic units ($\text{cm}^3$).

When the Base Area Is Given

Sometimes a problem gives you the base area directly, so you skip Step 1.

Worked Example

A prism has base area $B = 18\text{ m}^2$ and height $h = 7\text{ m}$.

$V = B \times h = 18 \times 7 = 126 \text{ m}^3$

You did not need the length and width separately — the base area has already combined them.

⚠️ Unit check: The base area must be in square units and the height in matching length units. Here $\text{m}^2 \times \text{m} = \text{m}^3$. ✓

Concept Check 🎯

Two-Step Volume 🧮

First find the base area $B = l \times w$, then $V = B \times h$. Enter the requested number.

1) $l = 9$, $w = 2$. Base area $B = \,?$ 2) Using that base with $h = 5$: $V = \,?$ 3) A prism with base area $B = 20$ and height $h = 6$: $V = \,?$

Match the Units 🔽

Choose the correct unit for each quantity in a problem measured in feet.

📦 Volume of Rectangular Prisms

Part 4 of 5 — Word Problems & Missing Sides

🔑 Working backward: If you know the volume and two of the three sides, you can find the missing side by dividing — because volume is built from multiplication, you undo it with division.

Finding a Missing Side

Start from $V = l \times w \times h$. To find one side, divide the volume by the product of the other two:

$h = \frac{V}{l \times w}$

Worked Example

A box has volume $V = 120\text{ cm}^3$, length $l = 5\text{ cm}$, and width $w = 4\text{ cm}$. Find the height.

Step 1 — Base area: $l \times w = 5 \times 4 = 20\text{ cm}^2$.

Step 2 — Divide: $h = \frac{120}{20} = 6 \text{ cm}$

✅ Check: $5 \times 4 \times 6 = 120$ ✓. The height is $6\text{ cm}$.

Concept Check 🎯

A Real-World Word Problem

A moving company uses boxes that are $2\text{ ft}$ long, $2\text{ ft}$ wide, and $3\text{ ft}$ tall. How many cubic feet does one box hold?

$V = 2 \times 2 \times 3 = 12 \text{ ft}^3$

Follow-up: If a truck can hold $600\text{ ft}^3$ of cargo, how many of these boxes fit?

$600 \div 12 = 50 \text{ boxes}$

💡 Strategy for word problems: (1) pull out $l$, $w$, $h$; (2) write $V = lwh$; (3) multiply carefully; (4) attach the cubic unit; (5) re-read the question to make sure you answered what was asked.

Word-Problem Practice 🧮

Enter each answer as a number.

1) A drawer is $30\text{ in}$ long, $20\text{ in}$ wide, and $10\text{ in}$ deep. Volume $= \,?\ \text{in}^3$ 2) A box has volume $48\text{ cm}^3$, length $6\text{ cm}$, width $2\text{ cm}$. Height $= \,?\ \text{cm}$ 3) A crate holds $72\text{ ft}^3$. If each small box is $9\text{ ft}^3$, how many boxes fit? $= \,?$

Set Up the Problem 🔽

A bin has volume $V = 36\text{ cm}^3$, length $3\text{ cm}$, and width $3\text{ cm}$. You want to find the height.

📦 Volume of Rectangular Prisms

Part 5 of 5 — Cubes, Mixed Practice & Mastery Check

You can now (1) picture volume as stacked unit cubes, (2) use $V = lwh$, (3) use $V = B \times h$, and (4) find a missing side. Let's add cubes and put it all together.

The Volume of a Cube

A cube has all sides equal, so if each side is $s$:

$V = s \times s \times s = s^3$

Worked Example: $s = 4\text{ cm}$

$V = 4^3 = 4 \times 4 \times 4 = 64 \text{ cm}^3$

Quick Reference

⚠️ Don't confuse $s^3$ with $3s$. $4^3 = 64$, but $3 \times 4 = 12$. The exponent means multiply $s$ by itself three times.

Cube Volumes 🧮

Use $V = s^3$. Enter the volume as a number.

1) $s = 2$. $V = \,?$ 2) $s = 5$. $V = \,?$ 3) $s = 10$. $V = \,?$

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.