A rectangular brick has dimensions 10 cm × 4 cm × 3 cm. What is the total surface area of the brick in square centimetres?

Introduction / Context:
This straightforward geometry question asks for the total surface area of a rectangular brick. A brick is essentially a cuboid, and its surface area can be found using the standard formula that involves the sum of products of its side lengths. The problem checks whether you remember the surface area formula for a cuboid and can apply it correctly to given dimensions in centimetres.

Given Data / Assumptions:

• Length of the brick, l = 10 cm.
• Breadth (width) of the brick, b = 4 cm.
• Height (thickness) of the brick, h = 3 cm.
• The brick has six rectangular faces, as a cuboid.

Concept / Approach:
The total surface area (TSA) of a cuboid with dimensions l, b and h is given by:
TSA = 2 * (l * b + b * h + l * h) This formula accounts for each pair of opposite faces: two faces of size l × b, two of size b × h, and two of size l × h. We substitute the given values into this formula to obtain the surface area in square centimetres.

Step-by-Step Solution:
Compute l * b = 10 * 4 = 40. Compute b * h = 4 * 3 = 12. Compute l * h = 10 * 3 = 30. Sum of products = 40 + 12 + 30 = 82. TSA = 2 * (l * b + b * h + l * h) = 2 * 82. TSA = 164 square centimetres.

Verification / Alternative check:
We can verify by explicitly accounting for each face: two faces of 40 cm^2 (total 80 cm^2), two faces of 12 cm^2 (total 24 cm^2), and two faces of 30 cm^2 (total 60 cm^2). Adding these gives 80 + 24 + 60 = 164 cm^2, which matches the formula-based calculation, confirming that the total surface area is correct.

Why Other Options Are Wrong:
84 sq.cm would result from taking only one of each face instead of both. 124 sq.cm and 180 sq.cm correspond to incorrect sums or misapplied formulas. 200 sq.cm would require larger face products than those actually present in this brick. None of these values match both the formula and the detailed face-by-face calculation.

Common Pitfalls:
A common mistake is to forget that each face occurs twice and therefore forget to multiply by 2 at the end. Another error is to confuse volume and surface area, mistakenly multiplying all three dimensions instead of using the surface area formula. Finally, incorrect arithmetic when adding 40, 12 and 30 also leads to wrong results.

Final Answer:
The total surface area of the brick is 164 sq.cm.