Surface Area of a Cuboid

Statement

The surface area of a cuboid is the total area of all six rectangular faces. A cuboid has three dimensions: length \(l\), width \(w\), and height \(h\). Each pair of opposite faces is congruent. The formula for the surface area is:

\[ S = 2(lw + lh + wh) \]

This formula adds up the areas of all the faces and doubles them, since opposite faces are equal.

Why it’s true

• The cuboid has three pairs of opposite faces: top/bottom, front/back, and left/right.
• Top and bottom each have area \(lw\); front and back each have area \(lh\); left and right each have area \(wh\).
• Total area = \(2lw + 2lh + 2wh = 2(lw + lh + wh)\).

Recipe (how to use it)

1. Identify the length, width, and height of the cuboid.
2. Multiply length × width, length × height, and width × height.
3. Add these three results together.
4. Multiply the sum by 2.
5. Write the final answer with square units (cm², m², etc.).

Spotting it

You use this formula whenever a 3D cuboid’s dimensions are given, and you need to find the total area to cover its surface (like wrapping paper or paint needed).

Common pairings

• Volume of a cuboid: \(V = l \times w \times h\).
• Area of rectangles and squares (building blocks for each face).

Mini examples

1. Given: \(l=5\), \(w=3\), \(h=2\). Find: \(S\). Answer: \(2(5\times3 + 5\times2 + 3\times2) = 62\).
2. Given: \(l=10\), \(w=4\), \(h=6\). Find: \(S\). Answer: \(2(40+60+24) = 248\).

Pitfalls

• Forgetting to double the sum of the face areas.
• Mixing up units (e.g., adding cm² with m²).
• Confusing surface area with volume.

Exam strategy

• Always label units clearly.
• Write down each multiplication separately to avoid mistakes.
• Check if the cuboid is actually a cube (all sides equal) — shortcut: \(S = 6a^2\).

Summary

The surface area of a cuboid measures the total covering area of all six rectangular faces. Use the formula \(S = 2(lw + lh + wh)\), being careful with units and remembering to multiply by 2. This formula is practical for real-world tasks like packaging, construction, or painting, and is a key GCSE skill.