Volume - Fifth Grade

Complete Notes & Formulas

What is Volume?

Volume is the amount of three-dimensional space that an object occupies. It measures how much space is inside a 3D shape.

Volume = Space Inside a 3D Shape

Measured in Cubic Units

Key Points About Volume

1. Volume is THREE-DIMENSIONAL

Measured in cubic units: cm³, m³, ft³, in³, etc.

2. Unit Cube

A cube with sides of 1 unit each has a volume of 1 cubic unit

3. Different from Area

Area = flat surface (2D), Volume = 3D space

4. Think of it as filling

How many unit cubes can fit inside?

1. Volume of Irregular Figures Made of Unit Cubes

What is an Irregular Figure?

An irregular figure is a 3D shape that is not a simple rectangular prism or cube. It's made by stacking unit cubes in different arrangements.

How to Find Volume

Volume = Count All Unit Cubes

Count every cube in the figure, including hidden ones!

Steps to Count Unit Cubes

Step 1: Count the visible cubes in the front

Step 2: Count the cubes on the sides (if visible)

Step 3: Count any hidden cubes behind or underneath

Step 4: Add all cubes together

Example

Problem: Find the volume of an L-shaped figure made of unit cubes.

Visual (L-shape):

Bottom layer: 5 cubes

Second layer: 3 cubes

Top layer: 2 cubes

Count all cubes:

Total = 5 + 3 + 2 = 10 unit cubes

Answer: 10 cubic units

Tip: Look carefully for hidden cubes! Count layer by layer to avoid mistakes.

2-4. Volume of Rectangular Prisms Made of Unit Cubes

What is a Rectangular Prism?

A rectangular prism is a 3D shape with 6 rectangular faces. It looks like a box.

Three Ways to Find Volume

Method 1: Counting Unit Cubes

Count all the unit cubes that fill the prism

Method 2: Using Expressions (Layers)

Volume = Cubes in Base Layer × Number of Layers

V = (length × width) × height

Method 3: Using Formula

V = l × w × h

V = B × h

where B = area of base (l × w)

Example 1: Counting Method

Problem: A rectangular prism is 4 cubes long, 3 cubes wide, and 2 cubes tall.

Bottom layer:

4 × 3 = 12 cubes

Number of layers: 2

Total volume:

12 × 2 = 24 cubic units

Answer: 24 cubic units

Example 2: Word Problem

Problem: A storage box is made of unit cubes. It is 5 units long, 4 units wide, and 3 units high. How many unit cubes fit inside?

V = l × w × h

V = 5 × 4 × 3

V = 60 unit cubes

Answer: 60 unit cubes can fit inside

5-6. Volume of Cubes and Rectangular Prisms

A. Volume of a Cube

A cube is a special rectangular prism where all edges are equal.

Volume of Cube = side³

V = s × s × s = s³

Example 1: Cube

Problem: Find the volume of a cube with side length 5 cm.

V = s³

V = 5³

V = 5 × 5 × 5

V = 125 cm³

Answer: 125 cubic centimeters

B. Volume of a Rectangular Prism

V = length × width × height

V = l × w × h

Example 2: Rectangular Prism

Problem: Find the volume of a rectangular prism: length = 8 m, width = 6 m, height = 4 m.

V = l × w × h

V = 8 × 6 × 4

V = 192 m³

Answer: 192 cubic meters

Word Problem Example

Problem: A swimming pool is 10 m long, 5 m wide, and 2 m deep. How much water can it hold?

V = l × w × h

V = 10 × 5 × 2

V = 100 m³

Answer: 100 cubic meters of water

7-8. Compare Volumes & Multi-Step Word Problems

Comparing Volumes

To compare volumes, calculate the volume of each prism and determine which is larger, smaller, or if they are equal.

Example 1: Compare Two Prisms

Problem: Which has a greater volume?

Prism A: 5 cm × 4 cm × 6 cm

Prism B: 8 cm × 3 cm × 5 cm

Prism A:

V = 5 × 4 × 6 = 120 cm³

Prism B:

V = 8 × 3 × 5 = 120 cm³

Answer: Both prisms have equal volume (120 cm³)

Multi-Step Word Problems

Example 2: Multi-Step Problem

Problem: A large box is 12 in × 10 in × 8 in. A smaller box is 6 in × 5 in × 4 in. How much more volume does the large box have?

Step 1: Find volume of large box

V₁ = 12 × 10 × 8 = 960 in³

Step 2: Find volume of small box

V₂ = 6 × 5 × 4 = 120 in³

Step 3: Find the difference

Difference = 960 − 120 = 840 in³

Answer: The large box has 840 in³ more volume

9. Volume of Compound Figures

What is a Compound Figure?

A compound figure (or composite figure) is a 3D shape made by combining two or more simple 3D shapes.

How to Find Volume

Step 1: Break the compound figure into simple rectangular prisms

Step 2: Find the dimensions of each prism

Step 3: Calculate the volume of each prism

Step 4: Add all the volumes together

Total Volume = Volume₁ + Volume₂ + Volume₃ + ...

Example: L-Shaped Building

Problem: An L-shaped building is made of two rectangular prisms.

Prism 1: 10 m × 4 m × 3 m

Prism 2: 6 m × 4 m × 3 m

Prism 1:

V₁ = 10 × 4 × 3 = 120 m³

Prism 2:

V₂ = 6 × 4 × 3 = 72 m³

Total Volume:

V = 120 + 72 = 192 m³

Answer: 192 cubic meters

10. Volume with Decimal Side Lengths

Same Formula, Decimal Numbers

The formula stays the same! Just multiply decimal numbers carefully.

V = l × w × h

(works with decimals too!)

Example 1: Decimal Dimensions

Problem: Find the volume: length = 5.5 cm, width = 3.2 cm, height = 4 cm.

V = l × w × h

V = 5.5 × 3.2 × 4

V = 17.6 × 4

V = 70.4 cm³

Answer: 70.4 cubic centimeters

Example 2: Cube with Decimal Side

Problem: A cube has sides of 2.5 meters. Find its volume.

V = s³

V = 2.5³

V = 2.5 × 2.5 × 2.5

V = 15.625 m³

Answer: 15.625 cubic meters

11. Compare and Apply Cubic Units

Common Cubic Units

Unit	Symbol	Used For
Cubic millimeter	mm³	Very small objects
Cubic centimeter	cm³	Small objects, medicine
Cubic meter	m³	Rooms, buildings, pools
Cubic inch	in³	Small objects (US)
Cubic foot	ft³	Rooms, storage (US)

Converting Cubic Units

Important Conversions:

1 m = 100 cm → 1 m³ = 1,000,000 cm³

1 ft = 12 in → 1 ft³ = 1,728 in³

1 cm = 10 mm → 1 cm³ = 1,000 mm³

When to Use Which Unit

cm³: Box of tissues, small container, dice

m³: Room, swimming pool, truck cargo space

in³: Juice box, small package

ft³: Refrigerator, closet, storage unit

Quick Reference: Volume Formulas

Shape	Volume Formula	Example
Cube	V = s³	5³ = 125
Rectangular Prism	V = l × w × h	8 × 6 × 4 = 192
Unit Cubes	Count all cubes	24 cubes = 24
Compound Figure	V₁ + V₂ + ...	120 + 72 = 192