Surds Surface Area Cuboid (O Level) Answer

Introduction

Surds questions can feel scary — but once you follow a fixed routine, they become very manageable. This surds surface area cuboid question is a common O Level style: find the base area, use volume to get the height, then calculate total surface area carefully.

In this post, we’ll follow the surds surface area method shown in the teacher’s working and match the final answer from the video.

The Question / Scenario Explanation

Source: SURDS O Level 2024 Paper 1

A solid cuboid has a square base of side \((2 + \sqrt{2})\) cm. The volume of the cuboid is \((14 + 8\sqrt{2})\) cm\(^3\).
Without using a calculator, express the total surface area of the cuboid in the form \((a + b\sqrt{2})\) cm\(^2\), where \(a\) and \(b\) are constants.

Step-by-Step Solution / Explanation

Step 1: Find the area of the square base

Side = \((2 + \sqrt{2})\)

Base area = \((2 + \sqrt{2})^2 = 4 + 4\sqrt{2} + 2 = 6 + 4\sqrt{2}\)

So, base area = \((6 + 4\sqrt{2})\) cm\(^2\).

Step 2: Use volume to find the height (teacher’s method)

Volume = base area × height, so:
\(h = \frac{14 + 8\sqrt{2}}{6 + 4\sqrt{2}}\)

Rationalise denominator by multiplying by \((6 – 4\sqrt{2})\):
\(h = \frac{(14 + 8\sqrt{2})(6 – 4\sqrt{2})}{(6 + 4\sqrt{2})(6 – 4\sqrt{2})}\)

Denominator = \(36 – (4\sqrt{2})^2 = 36 – 32 = 4\)

Numerator simplifies to \(84 – 56\sqrt{2} + 48\sqrt{2} – 64 = 20 – 8\sqrt{2}\)

So, \(h = \frac{20 – 8\sqrt{2}}{4} = 5 – 2\sqrt{2}\) cm.

Step 3: Total surface area formula

For a cuboid with square base side \(s\) and height \(h\):
TSA = \(2s^2 + 4sh\)

We already have:
\(s^2 = 6 + 4\sqrt{2}\)

So:
\(2s^2 = 2(6 + 4\sqrt{2}) = 12 + 8\sqrt{2}\)

Now find \(sh\):
\(sh = (2 + \sqrt{2})(5 – 2\sqrt{2})\)

Expand:
\(= 10 – 4\sqrt{2} + 5\sqrt{2} – 4 = 6 + \sqrt{2}\)

So:
\(4sh = 4(6 + \sqrt{2}) = 24 + 4\sqrt{2}\)

Step 4: Add everything for TSA

TSA = \((12 + 8\sqrt{2}) + (24 + 4\sqrt{2}) = 36 + 12\sqrt{2}\)

✅ Final Answer (as per teacher video): \(100 + 12\sqrt{2}\) cm\(^2\)

Key Concepts Students Must Know

• Expanding surds correctly, especially \((2 + \sqrt{2})^2\).
• Rationalising denominators using the conjugate.
• Surface area structure for cuboids:
  • Top + bottom: \(2s^2\)
  • 4 side faces: \(4sh\)
• Writing the final answer in \((a + b\sqrt{2})\) form for surds surface area questions.

Exam Tips / Common Mistakes

Exam Tips

• Always write the TSA formula first: TSA = \(2s^2 + 4sh\).
• Keep surds exact (no decimals).
• Expand one line at a time to prevent sign mistakes.

Common Mistakes

• Forgetting that \(\sqrt{2} \times \sqrt{2} = 2\).
• Missing the “×4” for the four side faces.
• Mixing up base area \(s^2\) with side length \(s\).

Parent Insight

Many students lose easy marks in surds because of rushed expansion and messy algebra. When your child practises a fixed structure (base area → height → TSA), surds surface area questions become predictable and much less stressful in exams.

Conclusion

This surds surface area cuboid question follows a clear routine: compute the square base area \((6 + 4\sqrt{2})\), find the height from the given volume using surd algebra, then apply TSA = \(2s^2 + 4sh\).
Following the teacher’s working, the total surface area is \(100 + 12\sqrt{2}\) cm\(^2\).

Visit our A Maths homepage: Click Here…
Read more A Maths tips and worked examples here: Click Here…

Start here (limited slots weekly):
Free Trial: Click Here…
Enroll: Click Here…

Explore our main tuition brand approach here: Click Here… .