Surface Area of Rectangular Prisms

What is Surface Area?

Surface area is the total area of ALL the outer surfaces of a 3D shape.

Think of it as: How much wrapping paper would you need to cover the entire box?

Measured in square units:

• cm² (square centimeters)
• m² (square meters)
• ft² (square feet)
• in² (square inches)

Rectangular Prism Faces

A rectangular prism has 6 faces (all rectangles):

• Top and bottom (2 faces)
• Front and back (2 faces)
• Left and right sides (2 faces)

Key insight: Opposite faces are identical!

Surface Area Formula

Method 1: Add all 6 faces

Find area of each face, then add them all.

Method 2: Formula

SA = 2lw + 2lh + 2wh

Where:

• l = length
• w = width
• h = height

This formula adds the three different pairs of faces!

Method 1: Add Each Face

Example: Box 4 cm × 3 cm × 2 cm

Step 1: Identify dimensions

• Length = 4 cm
• Width = 3 cm
• Height = 2 cm

Step 2: Find area of each pair of faces

Top & bottom: 4 × 3 = 12 cm² each

• Total: 2 × 12 = 24 cm²

Front & back: 4 × 2 = 8 cm² each

• Total: 2 × 8 = 16 cm²

Left & right: 3 × 2 = 6 cm² each

• Total: 2 × 6 = 12 cm²

Step 3: Add all faces

• 24 + 16 + 12 = 52 cm²

Answer: 52 cm²

Method 2: Using the Formula

Example: Same Box (4 × 3 × 2)

Formula: SA = 2lw + 2lh + 2wh

Substitute:

• l = 4, w = 3, h = 2
• SA = 2(4×3) + 2(4×2) + 2(3×2)
• SA = 2(12) + 2(8) + 2(6)
• SA = 24 + 16 + 12
• SA = 52 cm²

Answer: 52 cm²

Both methods give the same answer! Use whichever you prefer.

Alternative Formula

SA = 2(lw + lh + wh)

This factors out the 2:

Example: 4 × 3 × 2 box

• SA = 2(12 + 8 + 6)
• SA = 2(26)
• SA = 52 cm²

Surface Area of a Cube

A cube has 6 identical square faces.

Formula: SA = 6s²

Where s = side length

Example: Cube with side 5 cm

Formula: SA = 6s²

Calculate: SA = 6 × 5² = 6 × 25 = 150 cm²

Answer: 150 cm²

Why? Each face is 5 × 5 = 25 cm², and there are 6 faces.

Finding Missing Dimensions

Example: Find Height

Surface area = 94 m², Length = 5 m, Width = 3 m, Height = ?

Formula: SA = 2lw + 2lh + 2wh

Substitute:

94 = 2(5×3) + 2(5×h) + 2(3×h)
94 = 30 + 10h + 6h
94 = 30 + 16h
64 = 16h
h = 4 m

Surface Area vs. Volume

IMPORTANT: These measure different things!

Example: 3 × 4 × 5 box

Surface Area:

• SA = 2(3×4 + 3×5 + 4×5)
• SA = 2(12 + 15 + 20)
• SA = 94 units²

Volume:

• V = 3 × 4 × 5
• V = 60 units³

Different numbers, different units!

Real-World Applications

Painting a room: Calculate wall area

• 4 walls to paint (ignore floor and ceiling)
• Length 12 ft, width 10 ft, height 8 ft
• Front & back: 2(12×8) = 192 ft²
• Sides: 2(10×8) = 160 ft²
• Total wall area: 352 ft²

Gift wrapping: Box 15 in × 10 in × 8 in

• SA = 2(15×10 + 15×8 + 10×8)
• SA = 2(150 + 120 + 80)
• SA = 700 in²

Building materials: Cover a shipping container

• Dimensions 20 ft × 8 ft × 8 ft
• Find SA to determine material needed

Cardboard box: Manufacturing cost depends on surface area

• More surface area = more cardboard = higher cost

Nets of Rectangular Prisms

A net is a 2D pattern that folds into a 3D shape.

Strategy: Draw the net, find area of each rectangle, add them up.

This is the same as Method 1!

Practice