Volume Formulas and Composite Figures

Multiplying and Adding to Find Volume

In this lesson:

• Connect layer counting to the formula
• Use two equivalent volume formulas
• Decompose composite figures to find total volume

Learning Objectives for This Lesson

1. Connect layer counting to
2. Apply the formula to rectangular prisms
3. Explain why equals
4. Recognize that volume is additive
5. Decompose composite figures for total volume

Quick Review: Counting Unit Cubes

From your earlier lessons (5.MD.C.3 and 5.MD.C.4):

• Volume measures the space inside a solid figure
• We measure volume by packing unit cubes with no gaps
• Units: cubic centimeters, cubic inches, cubic feet

Today's big question: Can we find volume without counting every cube?

Building a Prism Layer by Layer

Each layer has the same number of cubes — just multiply!

Layers Reveal a Multiplication Pattern

Total = layer × layers

The Volume Formula: A Counting Shortcut

• Length × Width = cubes in one layer (area of the base)
• × Height = total cubes in all layers

Every time you use this formula, you are counting cubes!

Your Turn: Apply the Volume Formula

A prism: 5 units long, 4 wide, 2 tall.

Step 1: Cubes in one layer? 5 × 4 = ?

Step 2: Number of layers? 2

Step 3: Total volume = ? cubic units

Try it, then advance for the answer.

Check Your Answer and Discover Something

What does 5 × 4 give us on its own?

• The area of the base: square units
• So

Two Equivalent Formulas for Finding Volume

Formula 1:

Formula 2: where = base area

Group into :

is the area of the base.

Same Prism, Three Orientations, Same Volume

No matter which face is the "base," the volume is always 72 cubic units.

Example: Base Area Given Directly

A prism has a base area of 15 square inches and a height of 7 inches. What is its volume?

Solution:

We didn't need and separately — was enough!

Quick Check: Find the Base Area

What is for a prism with a 6 cm by 3 cm base?

If the height is 5 cm, what is the volume?

Think, then advance for the answer.

From Formulas to Real-World Problems

You now have two powerful volume formulas:

Next: Let's use them to solve real-world problems — finding volume, finding a missing dimension, and comparing two boxes.

Problem: How Much Fits Inside?

A toy chest is 4 ft long, 2 ft wide, and 2 ft tall. What is its volume?

Solution:

The toy chest holds 16 cubic feet of toys.

Working Backward to Find Missing Dimensions

A garden bed holds 48 cubic feet. It is 8 ft long and 2 ft wide.

The soil is 3 feet deep.

Problem: Which Tank Holds More?

Tank A: 12 × 8 × 10 inches

Tank B: 15 × 6 × 9 inches

Tank A holds 150 cubic inches more than Tank B.

Practice: Find and Compare Prism Volumes

Label all answers with cubic units.

1. Shipping box: 10 × 6 × 4 in. Volume?
2. Container: 120 cu cm, base 10 × 3 cm. Height?
3. Box A: 7 × 5 × 4. Box B: 6 × 8 × 3. Greater?

Try all three, then advance.

Answers: Volumes and Missing Dimensions Revealed

1. cubic inches
2. , so cm
3. ; — Box B wins by 4 cubic units

Always label with cubic units!

What About Shapes Like This?

Can we use on an L-shaped figure?

No — it isn't a single rectangular prism.

But we can break it into two rectangular prisms, find each volume, and add.

This works because volume is additive.

Volume Is Additive: The L-Shape

• Break the L-shape into two non-overlapping prisms
• Find each volume separately, then add
• The two prisms must not overlap — no double counting!

Decomposition A: Splitting with a Horizontal Cut

Prism 1 (bottom): 6 × 2 × 2 = 24 cubic cm

Prism 2 (right): 3 × 2 × 2 = 12 cubic cm

Decomposition B: Same Shape, Different Cut

Prism 1 (left): 3 × 4 × 2 = 24 cubic cm

Prism 2 (right): 3 × 2 × 2 = 12 cubic cm

Same total! Different cuts always give the same volume.

Real-World Problem: A Concrete Patio

Longer section: 10 × 4 × 1 ft = 40 cubic ft

Shorter section: 6 × 3 × 1 ft = 18 cubic ft

Your Turn: Decompose This T-Shape

• Top bar: 8 × 2 × 3 cm
• Vertical stem: 2 × 4 × 3 cm

Step 1: Find the volume of the top bar.

Step 2: Find the volume of the stem.

Step 3: Add for the total.

Try it, then advance for the answer.

Answer: Computing the T-Shape Total Volume

Top bar: cubic cm

Stem: cubic cm

Circle each prism's dimensions before multiplying!

Key Takeaways from Today's Lesson

✓ counts cubes by multiplying

✓ groups base area into

✓ Composite: decompose, compute, add

Area is flat. Volume is filled — don't forget .

Match dimensions to their own prism first.

What Comes Next in Grade Six

You've learned:

• Volume formulas for rectangular prisms
• Additive volume for composite figures

Coming up:

• Volume with fractional edge lengths (6.G.A.2)
• Surface area of 3D figures (6.G.A.4)
• extends to new base shapes!