Volume of Rectangular Prisms

📦 Volume of Rectangular Prisms

Part 1 of 5 — Concept Introduction

When you measure how much flat space a shape covers, you find its area. But the real world is not flat — boxes, swimming pools, and storage units take up space in three directions at once.

Volume measures how much space is inside a 3D shape. A handy way to picture it: "How many unit cubes would fit inside?"

A Rectangular Prism

A rectangular prism is a box shape with 6 rectangular faces. A cereal box, a brick, and a shoebox are all rectangular prisms. It has three measurements:

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

Cubic Units

Volume is always measured in cubic units, because we are counting unit cubes. A unit cube is $1$ unit long, $1$ unit wide, and $1$ unit tall.

Measurement	Directions	Example Units
Length	1 direction	$cm$, $in$
Area	2 directions	$cm^2$, $in^2$
Volume	3 directions	$cm^3$, $in^3$

The little raised $3$ in $cm^3$ means "cubic." It reminds us we multiplied three measurements: length, width, and height. 🧊

The Volume Formula

To find the volume of any rectangular prism, multiply all three measurements together:

$V = l \times w \times h$

There is a second, very useful way to think about it. The bottom of the box is a rectangle called the base. Its area is $l \times w$. Then you stack that base upward by the height:

$V = (\text{Area of base}) \times h$

Both ideas give the exact same answer — they are just two ways of seeing the same multiplication.

A Special Case: Cubes

A cube is a rectangular prism where all three sides are equal. If each side has length $s$, then:

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

So a cube with side $5\,cm$ has volume $5^3 = 5 \times 5 \times 5 = 125\,cm^3$.

A Worked Example

Let's find the volume of a box that is $7\,cm$ long, $4\,cm$ wide, and $3\,cm$ tall.

Step 1 — Write the formula:

$V = l \times w \times h$

Step 2 — Put in the numbers:

$V = 7 \times 4 \times 3$

Step 3 — Multiply, one piece at a time:

$V = 28 \times 3 = 84$

Step 4 — Add the cubic units:

$V = 84\,cm^3$

So $84$ unit cubes would fit inside the box. Notice the unit is $cm^3$ (cubic), not $cm$ or $cm^2$. ✅

Concept Check 🎯

Make sure you understand what volume measures and which units we use.

📝 Worked Examples

Part 2 of 5 — Worked Examples

Let's slow down and work through two problems one step at a time.

Example 1: A Bigger Box

Find the volume of a box that is $6\,in$ long, $5\,in$ wide, and $4\,in$ tall.

Step 1 — Formula: $V = l \times w \times h$

Step 2 — Substitute: $V = 6 \times 5 \times 4$

Step 3 — Multiply the first two: $6 \times 5 = 30$

Step 4 — Multiply by the last one: $30 \times 4 = 120$

Step 5 — Add units: $V = 120\,in^3$ ✅

💡 Tip: It does not matter which two numbers you multiply first. $6 \times 5 \times 4$, $4 \times 6 \times 5$, and $5 \times 4 \times 6$ all equal $120$.

Example 2: Using the Base

Find the volume of an aquarium that is $12\,cm$ long, $5\,cm$ wide, and $8\,cm$ tall — this time using the base method.

Step 1 — Find the area of the base (the bottom rectangle):

$\text{Area of base} = l \times w = 12 \times 5 = 60\,cm^2$

Step 2 — Multiply the base area by the height:

$V = (\text{Area of base}) \times h = 60 \times 8 = 480$

Step 3 — Add units: $V = 480\,cm^3$ ✅

This is the same answer you would get from $12 \times 5 \times 8$. Two paths, one answer! 🎉

Your Turn 🧮

A storage bin is $9\,cm$ long, $4\,cm$ wide, and $6\,cm$ tall. Find its volume step by step. Type only the number in each box.

1. First, multiply length and width: $9 \times 4 = ?$ (This is the area of the base.)

2. Now multiply that result by the height: (your answer) $\times 6 = ?$ (This is the volume in $cm^3$.)

🧭 Guided Practice

Part 3 of 5 — Guided Practice

Use $V = l \times w \times h$ on each problem. Remember to add cubic units.

Fill In the Steps 🔍

A prism is $7\,ft$ long, $3\,ft$ wide, and $2\,ft$ tall. Complete the two steps to find its volume.

🌍 Application & Word Problems

Part 4 of 5 — Real-World Problems

Volume shows up everywhere in real life. We use it to figure out how much a container can hold, how much water fills a pool, or how much soil fills a garden box.

Aquarium Example 🐠

Jordan is filling an aquarium shaped like a rectangular prism. It is $50\,cm$ long, $30\,cm$ wide, and $40\,cm$ tall. How much water can it hold?

This question asks for the volume:

$V = l \times w \times h = 50 \times 30 \times 40$

Multiply two at a time:

$50 \times 30 = 1{,}500 \qquad 1{,}500 \times 40 = 60{,}000$

The aquarium holds $60{,}000\,cm^3$ of water! 🌊

Watch Your Words

Word problems often hide the measurements inside a sentence. Look for three numbers — the length, width, and height — and multiply them together. The answer is always in cubic units.

Word Problem Practice 🧮

A garden planter is $8\,ft$ long, $3\,ft$ wide, and $2\,ft$ tall. Find how much soil it holds, step by step. Type only the number in each box.

1. Multiply length and width: $8 \times 3 = ?$ (the base area)

2. Multiply that result by the height: (your answer) $\times 2 = ?$ (the volume in $ft^3$)

Real-World Check ✅

Read carefully and pick the volume.

🏆 Review & Challenge

Part 5 of 5 — Putting It All Together

You made it! Here is everything about volume in one quick summary.

Summary Table

Idea	What to Remember
What is volume?	The space inside a 3D shape
Formula	$V = l \times w \times h$
Base method	$V = (\text{Area of base}) \times h$
Cube	$V = s^3$ (all sides equal)
Units	Always cubic: $cm^3$, $m^3$, $in^3$, $ft^3$

Key Reminders 🔑

• Volume always uses three measurements multiplied together.
• The answer is always in cubic units (the little $3$ in $cm^3$).
• A cube is just a rectangular prism with all equal sides, so $V = s^3$.

Now try the final challenge to mix everything together!

Final Challenge 🎯

These problems mix together every idea from the lesson. Take your time!