Surface Area of a Cuboid –

Surface area of a cuboid is one of the most common mensuration questions on GCSE Maths papers at both Foundation and Higher tiers. You need to identify the three pairs of rectangular faces, apply the formula, and adapt it for variations like open-top boxes. This guide explains the formula, walks through worked examples at both levels, and provides practice questions.

What Is the Surface Area of a Cuboid?

The surface area of a 3D shape is the total area of all its faces. A cuboid has six rectangular faces that come in three identical pairs:

• Two faces of size length x width (top and bottom).
• Two faces of size length x height (front and back).
• Two faces of size width x height (left and right).

Key Formulas

In the open-top version, only one of the two lw faces is counted — the base but not the missing top.

Step-by-Step Method

1. Identify the length (l), width (w), and height (h) of the cuboid.
2. Calculate the area of each pair of faces: lw, lh, and wh.
3. Multiply each by 2 to account for both faces in the pair.
4. Add all three results together.
5. If the cuboid is open-top, subtract one lw face from the total.

Worked Example 1 — Foundation Level

Question: Find the surface area of a cuboid with length 8 cm, width 5 cm, and height 3 cm.

Working:

Step 1 — Area of top and bottom = 2 × (8 × 5) = 2 × 40 = 80 cm².

Step 2 — Area of front and back = 2 × (8 × 3) = 2 × 24 = 48 cm².

Step 3 — Area of left and right = 2 × (5 × 3) = 2 × 15 = 30 cm².

Step 4 — Total surface area = 80 + 48 + 30 = 158 cm².

Answer: The surface area is 158 cm².

Worked Example 2 — Higher Level

Question: A cuboid has a surface area of 214 cm². Its length is 7 cm and width is 5 cm. Find the height.

Working:

Step 1 — Use the formula: 2(lw + lh + wh) = 214.

Step 2 — Substitute l = 7 and w = 5: 2(35 + 7h + 5h) = 214.

Step 3 — Simplify: 2(35 + 12h) = 214.

Step 4 — Divide both sides by 2: 35 + 12h = 107.

Step 5 — Subtract 35: 12h = 72.

Step 6 — Divide by 12: h = 6 cm.

Answer: The height is 6 cm.

Worked Example 3 — Exam Style

Question: A metal tray is made from a rectangular sheet of metal by cutting 2 cm squares from each corner and folding up the sides. The original sheet is 20 cm by 14 cm. Find the surface area of the inside of the tray.

Working:

Step 1 — After cutting and folding, the base measures (20 − 4) × (14 − 4) = 16 × 10 cm. The height = 2 cm.

Step 2 — The tray is an open-top cuboid: l = 16, w = 10, h = 2.

Step 3 — Inside surface area = base + 4 sides = (16 × 10) + 2(16 × 2) + 2(10 × 2).

Step 4 — = 160 + 64 + 40 = 264 cm².

Answer: The inside surface area is 264 cm².

Common Mistakes

• Forgetting to multiply by 2. Each face appears twice in a cuboid — students often calculate lw + lh + wh and forget to double.
• Confusing surface area with volume. Surface area is measured in cm²; volume is measured in cm³. The formulas are completely different.
• Not adjusting for open-top boxes. If the top is missing, subtract one lw face from the standard formula.

Exam Tips

• Write out all three pair calculations separately before adding — this earns method marks even if you make an arithmetic error.
• When given a net, calculate the area of each face directly rather than trying to use the formula.
• In reverse problems (given SA, find a dimension), set up the equation carefully and solve step by step.
• Always include the correct units — cm² for area.

Practice Questions

Q1 (Foundation): Find the surface area of a cube with side length 6 cm.

Q2 (Foundation): A cuboid has dimensions 10 cm × 4 cm × 3 cm. Find the surface area.

Q3 (Higher): An open-top fish tank is 60 cm long, 30 cm wide, and 35 cm tall. How much glass is needed to make the tank?

Practise surface area questions with instant feedback free on GCSEMathsAI.

Related Topics

• Surface Area — surface area of other 3D shapes.
• Volume of 3D Shapes — volume calculations for cuboids and more.
• Nets of 3D Shapes — visualising faces using nets.

Summary

Surface area of a cuboid is one of the most accessible GCSE marks available. The formula SA = 2(lw + lh + wh) accounts for three pairs of rectangular faces. For open-top boxes, remove one lw face. Always calculate each pair separately to avoid errors, double-check that you have multiplied by 2, and use correct units. Reverse problems where you are given the surface area and must find a missing dimension require careful algebraic rearrangement.