Why Do We Need to Find Volume?
People have measured volume for thousands of years. Think about it: if you were building a house long ago, you needed to know how much stone or clay would fill the walls. If you were a baker, you needed to know how much flour fit inside a container. Volume tells us how much space a solid shape takes up.
Over time, people realized that many real objects are not simple boxes. A building might have an L-shape. A swimming pool might be deeper on one end. To figure out the volume of these shapes, we need a clever trick: break the shape into smaller, simpler pieces and then add the pieces together.
The big question this lesson answers is: How do you find the volume of a shape that is made of two rectangular prisms stuck together? Let's find out!
Core Ideas You Need to Know
Before we tackle composite figures, let's review a few important ideas. These are the building blocks you will use throughout this lesson.
Volume
Rectangular Prism
Volume Formula
Composite Figure
Volume Is Additive
See It: Splitting a Composite Figure
Let's look at an L-shaped figure. It's made of two rectangular prisms joined together. The diagram below shows how we can draw a line to split the shape into two simpler boxes. Each box has its own length, width, and height.
In the diagram above, Prism A is 4 cm long, 2 cm wide, and 3 cm tall. Prism B is 3 cm long, 2 cm wide, and 5 cm tall. Notice that the two prisms do not overlap — they share a face but they don't take up the same space. That means we can simply add the two volumes together.
The Math: Formulas You Will Use
There are only two formulas you need. The first one finds the volume of a single rectangular prism. The second one adds the parts together.
Let's use the shape from our diagram. Prism A: V₁ = 4 × 2 × 3 = 24 cm³. Prism B: V₂ = 3 × 2 × 5 = 30 cm³. Total volume = 24 + 30 = 54 cm³. Done!
Different Ways to Split the Same Shape
Here is a cool fact: you can sometimes split a composite figure in more than one way. You might cut it with a horizontal line or a vertical line. Either way, the total volume will be the same! The diagram below shows two different ways to divide a T-shaped figure.
No matter how you split the shape, the total volume stays the same. Pick the way that makes the numbers easiest for you. Sometimes a horizontal cut gives nicer numbers. Sometimes a vertical cut works better. You get to choose!
Worked Example: The Toy Castle
Maya is building a toy castle out of wooden blocks. The castle has a tall tower on top of a wide base. The base is 10 cm long, 6 cm wide, and 4 cm tall. The tower is 3 cm long, 3 cm wide, and 7 cm tall. It sits on top of the base without overlapping any part. What is the total volume of the castle?
Helpful Tips and Common Mistakes
Even though the idea is simple — find each volume and add — there are a few places where students sometimes trip up. The table below compares good habits with common mistakes.
| Good Habit ✅ | Common Mistake ❌ | Why It Matters |
|---|---|---|
| Label each prism's length, width, and height before multiplying. | Mixing up numbers between the two prisms. | Swapping a number gives the wrong volume for both parts. |
| Make sure the two parts do NOT overlap. | Counting a shared section in both prisms. | You'd be counting some cubes twice, making the total too big. |
| Use the same units for every measurement. | Mixing inches and centimeters in the same problem. | Different units give a meaningless answer. |
| Write cubic units (cm³, in³, ft³) in your answer. | Forgetting the ³ or writing just "cm". | Volume is measured in cubic units, not plain units. |
From Composite Boxes to Bigger Ideas
Right now you are working with two rectangular prisms stuck together. But the same "split and add" idea works for even more complex shapes! As you move into middle school and beyond, you'll use this strategy for prisms, cylinders, and other 3D figures.
| What You Know Now | What Comes Next |
|---|---|
| Volume of rectangular prisms: V = l × w × h | Volume of triangular prisms and cylinders |
| Splitting into 2 non-overlapping parts | Splitting into 3 or more parts |
| Adding volumes | Sometimes subtracting — like finding the volume of a hole cut out of a block |
| Simple whole-number dimensions | Decimal and fraction dimensions |
The big idea — volume is additive — will follow you all the way through math class. It's like a superpower that lets you handle any 3D shape, no matter how strange it looks!
Practice Problems
Try these five problems. They start easy and get harder. Show your work and remember to include cubic units!
Lesson Summary
In this lesson you learned that volume is additive. When a solid shape is made of two non-overlapping rectangular prisms, you can find the volume of each prism using V = l × w × h and then add them together to get the total volume. This works because the two parts fill different spaces — no cubes are counted twice.
You also learned that a composite figure can be split in more than one way and the answer will stay the same. Always label your dimensions carefully, use cubic units in your answer, and check that the two parts do not overlap. This skill helps you solve real-world problems like filling garden beds, packing boxes, and designing buildings!