Find Volume of Composite Figures - 5th Grade Math

SECTION 1

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.

3000 BC • Ancient Egypt

Egyptians measured the volume of stone blocks to build the pyramids. Each block was a rectangular prism!

300 BC • Greek Math

A mathematician named Euclid wrote rules for finding the volume of boxes and other 3D shapes.

1800s • Standard Units

Countries agreed on standard units like cubic centimeters and cubic feet so everyone could measure volume the same way.

Today • Composite Figures

Engineers, architects, and even video game designers split complex shapes into simpler boxes to find their total volume.

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!

SECTION 2

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

Volume is the amount of space inside a 3D shape. We measure it in cubic units like cubic centimeters (cm³) or cubic inches (in³).

Rectangular Prism

A rectangular prism is a 3D shape with 6 flat faces that are all rectangles. Think of a cereal box or a brick.

Volume Formula

To find the volume of a rectangular prism, multiply: length × width × height. That's it!

Composite Figure

A composite figure is a shape made by joining two or more simpler shapes together. The parts do not overlap.

Volume Is Additive

When two shapes don't overlap, you can add their volumes to get the total. This is the key idea of this lesson!

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KEY TAKEAWAY

Think of volume like filling a shape with sugar cubes. If you have an L-shaped candy box, you can split it into two smaller rectangular boxes. Count the cubes in each box, then add the two amounts together to get the total. That's what "volume is additive" means!

SECTION 3

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.

The yellow dashed line shows where the L-shape is split. Prism A is on the left and Prism B is on the right. Find each volume, then add!

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.

SECTION 4

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.

VOLUME OF A RECTANGULAR PRISM

V = l × w × h

V = volume, l = length, w = width, h = height. Multiply all three numbers together.

TOTAL VOLUME OF A COMPOSITE FIGURE

V_total = V₁ + V₂

V₁ = volume of the first prism, V₂ = volume of the second prism. Just add them!

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Remember!

The two prisms must not overlap. If they share space, you would be counting some cubes twice. Make sure your cut separates the shape into pieces that do not take up the same space.

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!

SECTION 5

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.

On the left, a horizontal cut gives two prisms. On the right, vertical cuts give three prisms. Both methods produce the same total volume.

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!

SECTION 6

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?

Finding the Volume of Maya's Castle

1
Step 1 — Identify the Two Prisms

The castle is made of two rectangular prisms that do not overlap. Prism 1 is the wide base. Prism 2 is the tall tower on top.
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Step 2 — Write Down Each Prism's Dimensions

Base (Prism 1): l = 10 cm, w = 6 cm, h = 4 cm. Tower (Prism 2): l = 3 cm, w = 3 cm, h = 7 cm.
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Step 3 — Find the Volume of Each Prism

V₁ = 10 × 6 × 4 = 240 cm³. V₂ = 3 × 3 × 7 = 63 cm³.

V₁ = 240 cm³, V₂ = 63 cm³
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Step 4 — Add the Volumes Together

V_total = V₁ + V₂ = 240 + 63 = 303 cm³.

Total volume = 303 cm³
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Step 5 — Check Your Answer

Does it make sense? The base alone holds 240 sugar cubes. The tower adds 63 more. 303 cubes total — yes, that's reasonable for a castle shape!

SECTION 7

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.

Tips vs. Mistakes

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PRO TIP

Before you add, do a quick check: does each part's volume seem right on its own? If a tiny tower has a bigger volume than a huge base, something is probably wrong. Trust your common sense!

SECTION 8

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

Now vs. Later

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!

SECTION 9

Practice Problems

Try these five problems. They start easy and get harder. Show your work and remember to include cubic units!

Question 1: CONCEPTUAL

Emma says, "I can find the volume of an L-shaped figure by finding the volume of one big rectangle that covers the whole shape." Is she right? Why or why not?

Show answer

Emma is not quite right. If she draws one big rectangle around the entire L-shape, that rectangle includes empty space that is not part of the L-shape. She would be counting volume that isn't really there. The correct method is to split the L-shape into two non-overlapping rectangular prisms, find each volume using V = l × w × h, and then add the two volumes together.

Question 2: BASIC CALCULATION

A composite figure is made of two rectangular prisms. Prism A is 5 in × 4 in × 3 in. Prism B is 2 in × 4 in × 6 in. They do not overlap. What is the total volume?

Show answer

First, find each volume. V₁ = 5 × 4 × 3 = 60 in³. V₂ = 2 × 4 × 6 = 48 in³. Then add: 60 + 48 = 108 in³. The total volume of the composite figure is 108 cubic inches. We used the formula V = l × w × h for each prism and then applied the additive rule.

Question 3: INTERMEDIATE

A step-shaped figure has a bottom prism that is 8 cm × 5 cm × 2 cm and a top prism that is 4 cm × 5 cm × 3 cm. The top prism sits on the left half of the bottom prism. What is the total volume? Could you split this shape a different way and still get the same answer?

Show answer

V₁ (bottom) = 8 × 5 × 2 = 80 cm³. V₂ (top) = 4 × 5 × 3 = 60 cm³. Total = 80 + 60 = 140 cm³. Yes, you could also split the shape with a vertical cut. The left part would be 4 × 5 × 5 = 100 cm³ and the right part would be 4 × 5 × 2 = 40 cm³. Adding gives 100 + 40 = 140 cm³ — the same answer!

Question 4: APPLIED

Carlos is filling a raised garden bed shaped like an L. The long part of the L is 6 ft × 2 ft × 1 ft. The short part is 3 ft × 2 ft × 1 ft. Soil costs $4 per cubic foot. How many cubic feet of soil does he need, and how much will it cost?

Show answer

Long part: V₁ = 6 × 2 × 1 = 12 ft³. Short part: V₂ = 3 × 2 × 1 = 6 ft³. Total volume = 12 + 6 = 18 ft³. At $4 per cubic foot, the cost is 18 × $4 = $72. Carlos needs 18 cubic feet of soil and will spend $72. This problem shows how volume helps us in real life — you need to know the space inside a shape to figure out how much material to buy.

Question 5: CRITICAL THINKING

Priya has a composite figure made of two rectangular prisms. The total volume is 200 cm³. Prism A has a volume of 120 cm³ and its dimensions are 10 cm × 4 cm × 3 cm. Prism B has a width of 4 cm and a height of 5 cm. What is the length of Prism B?

Show answer

Since volume is additive: V₂ = 200 − 120 = 80 cm³. We know V₂ = l × 4 × 5 = 20 × l. So 20 × l = 80, which means l = 80 ÷ 20 = 4 cm. The length of Prism B is 4 cm. This problem works the formula backward. Instead of multiplying to find volume, you used division to find a missing dimension. That's a great skill for solving trickier problems!

SUMMARY

Lesson Summary

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!