Volume of Rectangular Prisms

What is Volume?

Volume is the amount of space inside a 3D object.

Think of it as: How much water would fill the container?

Measured in cubic units:

• cm³ (cubic centimeters)
• m³ (cubic meters)
• ft³ (cubic feet)
• in³ (cubic inches)

What is a Rectangular Prism?

A rectangular prism is a 3D box shape with:

• 6 rectangular faces
• All angles are right angles (90°)

Examples:

• Cereal box
• Shoebox
• Brick
• Aquarium

Dimensions:

• Length (l): longest side of base
• Width (w): shorter side of base
• Height (h): how tall it is

Volume Formula

Volume = length × width × height

V = l × w × h or V = lwh

Why? Think of stacking unit cubes:

• First layer: l × w cubes
• Stack h layers
• Total: l × w × h cubes

Example 1: Basic Rectangular Prism

Length = 5 cm, Width = 3 cm, Height = 4 cm

Formula: V = lwh

Calculate: V = 5 × 3 × 4 = 60 cm³

Answer: 60 cm³

Example 2: Different Order

Height = 10 m, Width = 2 m, Length = 6 m

Order doesn't matter!

V = 6 × 2 × 10 = 120 m³

Answer: 120 m³

Example 3: Larger Numbers

Length = 12 ft, Width = 8 ft, Height = 5 ft

V = 12 × 8 × 5 = 480 ft³

Answer: 480 ft³

Volume of a Cube

A cube is a special rectangular prism where all sides are equal!

Formula: V = s³ (where s = side length)

This is the same as V = s × s × s

Example: Cube

Side length = 4 cm

V = 4³ = 4 × 4 × 4 = 64 cm³

Answer: 64 cm³

Finding Missing Dimensions

If you know the volume and two dimensions, you can find the third!

Example: Find Height

Volume = 180 cm³, Length = 9 cm, Width = 5 cm, Height = ?

Formula: V = lwh

Substitute: 180 = 9 × 5 × h

Solve:

180 = 45h
180 ÷ 45 = h
h = 4 cm

Example: Find Width

Volume = 240 m³, Length = 10 m, Height = 6 m, Width = ?

Substitute: 240 = 10 × w × 6

Solve:

240 = 60w
w = 4 m

Volume with Fractions

Example: Fractional Dimensions

Length = 4.5 cm, Width = 2 cm, Height = 3 cm

V = 4.5 × 2 × 3 = 27 cm³

Example: Fractions

Length = 1/2 m, Width = 1/4 m, Height = 3 m

V = (1/2) × (1/4) × 3 = 3/8 m³

Comparing Volume and Surface Area

Volume: Space inside (3D) → cubic units Surface Area: Total area of all faces (2D) → square units

These are completely different!

Example: Box with 2 × 3 × 4 dimensions

• Volume = 2 × 3 × 4 = 24 units³
• Surface Area = 2(2×3 + 2×4 + 3×4) = 52 units²

Composite Shapes

Break complex shapes into rectangular prisms!

Example: L-shaped building

• Split into two rectangular prisms
• Find volume of each
• Add them together

Example: Two-Part Shape

Part 1: 5 × 3 × 2 = 30 cm³ Part 2: 4 × 2 × 3 = 24 cm³

Total volume: 30 + 24 = 54 cm³

Real-World Applications

Aquarium: 24 in × 12 in × 16 in

• V = 24 × 12 × 16 = 4,608 in³
• How much water it holds

Moving box: 18 in × 12 in × 10 in

• V = 2,160 in³
• How much stuff fits

Swimming pool: 25 m × 10 m × 2 m

• V = 500 m³
• Amount of water needed

Refrigerator: 3 ft × 2.5 ft × 6 ft

• V = 45 ft³
• Storage space

Converting Units

Remember: When converting volume, use conversion factor THREE times!

Example: Convert 2 ft³ to in³

• 1 ft = 12 in
• 1 ft³ = 12 × 12 × 12 = 1,728 in³
• 2 ft³ = 2 × 1,728 = 3,456 in³

Capacity (Liquid Volume)

Sometimes volume is measured in:

• Liters (L) and milliliters (mL)
• Gallons, quarts, pints, cups

Conversion: 1 cm³ = 1 mL

• A prism with volume 1,000 cm³ holds 1,000 mL = 1 L

Practice