Volume | 5th Grade Math

Complete Notes on Volume - Grade 5

1. Volume of Irregular Figures Made of Unit Cubes

Volume is the amount of space inside a 3D shape, measured in cubic units (unit³).

To find the volume of an irregular figure made of unit cubes: Count the total number of cubes in the figure.

Example:

  • If a shape is made of 12 unit cubes, its volume is 12 cubic units.
  • Add all cubes together, no matter the arrangement.
✓ Label your answer with cubic units (cm³, m³, etc.)

2. Volume of Rectangular Prisms Made of Unit Cubes / Expressions

For rectangular prisms, use the formula:

\[ \text{Volume} = \text{length} \times \text{width} \times \text{height} \]

Or, \( V = l \times w \times h \)

Examples:

  • A prism 4 units long, 3 units wide, 2 units high:
    \( 4 \times 3 \times 2 = 24 \) cubic units
  • Given expression: \( 2 \times 5 \times 6 = 60 \) cubic units

3. Volume of Rectangular Prisms Made of Unit Cubes

Same formula: \( V = l \times w \times h \). Each cube has side 1 unit.

Find the area of one layer: length × width, then multiply by height.

Example:

  • Bottom layer (5 × 4): 20 cubes. 3 layers. Total: \( 20 \times 3 = 60 \) cubic units.

4. Volume of Rectangular Prisms & Cubes: Word Problems

Steps:

  • Draw or visualize the figure
  • Write down length, width, height
  • Multiply \( l \times w \times h \)
  • Write answer in cubic units

Example

  • A box 8 m × 5 m × 2 m: \( 8 \times 5 \times 2 = 80 \) m³

5. Volume of Cubes and Rectangular Prisms

Formulas:

  • CUBE: \( V = s^3 \), where \( s \) is the side length.
  • RECTANGULAR PRISM: \( V = l \times w \times h \)

Example:

  • Cube side = 4 cm. Volume = \( 4 \times 4 \times 4 = 64 \) cm³
  • Prism 3 in × 2 in × 5 in: \( 3 \times 2 \times 5 = 30 \) in³

6. Volume of Compound Figures

What are Compound Figures?

Compound (composite) figures are made of more than one prism/cube. Find the volume of each part, then add all together.

Example:

  • Part A is a box 6 × 2 × 1: \( 6 \times 2 \times 1 = 12 \) units³
    Part B is 3 × 2 × 2: \( 3 \times 2 \times 2 = 12 \) units³
    Total Volume = 12 + 12 = 24 units³
  • If subtracting a hole, do: whole volume − hole volume
✓ Break complex shapes into simple prisms or cubes.
✓ Add all volumes together.

7. Volume with Decimal Side Lengths

Just use the same formula. Multiply as usual, but be careful with decimals.

Example:

  • Box 2.5 m × 1.4 m × 3 m: \( 2.5 \times 1.4 \times 3 = 10.5 \) m³
✓ Line up decimal points when multiplying.

8. Compare and Apply Cubic Units

What are Cubic Units?

Volume is measured in cubic units (cm³, m³, in³, ft³, etc). Larger measurements mean more space.

Examples:

  • Cube with side 1 cm has volume 1 cm³. Cube with side 1 m = 1 m³ (which is much bigger!)
  • 2 boxes: One is 6 × 3 × 2 ft, the other is 3 × 3 × 3 ft. First volume = 36 ft³, second = 27 ft³.
    36 ft³ > 27 ft³
✓ Always check that units match.
✓ Compare by finding the volume in same units.

9. Multi-step Word Problems

How to Solve

  • Identify what the problem is asking
  • Take all known values
  • Do step-by-step calculation (may need to find areas first, then volumes)

Example:

  • A box 4 × 3 × 2 ft. How many 1 ft³ boxes fit? Volume = 4 × 3 × 2 = 24 ft³. 24 boxes of 1 ft³ fit inside.

📚 Quick Reference Formula Sheet

Cube

\[ V = s^3 \]   (where s is the side length)

Rectangular Prism

\[ V = l \times w \times h \]
where l = length, w = width, h = height

Compound Figures

\[ V_{total} = V_1 + V_2 + ... \]   (Add all volumes together)

Unit Cubes

\[ V = \text{Total number of unit cubes} \]