A square pyramid has a base that is a square of side 6 cm and a slant height of 5 cm. What is its slant (lateral) surface area (in square centimetres)?

Introduction / Context:
This question asks for the slant or lateral surface area of a square pyramid. The base is a square and each of the four lateral faces is a congruent isosceles triangle with given base and slant height. Lateral surface area is important in applications involving roof design, packaging, or any structure where the sides are triangular faces attached to a polygonal base.

Given Data / Assumptions:
• The base is a square with side length 6 cm.
• The slant height of the pyramid (height of each triangular face measured along its altitude) is 5 cm.
• We are asked for the slant surface area, meaning the total area of the four triangular faces, not including the base.

Concept / Approach:
Each lateral face is an isosceles triangle whose base equals the side of the square base and whose height equals the slant height of the pyramid. The area of one triangle is (1/2) * base * height. Since there are four such identical triangles, the total lateral area is 4 times the area of one triangle. There is no need to find the vertical height or total surface area, because the question specifically asks for slant surface area only.

Step-by-Step Solution:
Step 1: Base side length a = 6 cm. Step 2: Slant height l = 5 cm. Step 3: Area of one triangular face = (1/2) * base * height = (1/2) * 6 * 5. Step 4: Compute this area: (1/2) * 6 * 5 = 3 * 5 = 15 square centimetres. Step 5: There are four such triangular faces, so total slant area = 4 * 15 = 60 square centimetres.

Verification / Alternative check:
We can also use the general formula for the lateral surface area of a square pyramid: Lateral area = 2 * a * l, where a is the base side and l is the slant height. Substituting a = 6 and l = 5, we get 2 * 6 * 5 = 60 square centimetres. This matches the earlier calculation based on four triangular faces, so the answer is consistent and correct.

Why Other Options Are Wrong:
30 square centimetres corresponds to only two faces or half of the lateral surface area, so it is too small.
40 and 45 square centimetres do not correspond to any correct combination of triangular areas given the specified dimensions.
50 square centimetres is also not a multiple of the single triangle area of 15 square centimetres and therefore cannot represent the total lateral area of four identical faces.

Common Pitfalls:
A common mistake is to confuse the slant surface area with the total surface area and either add or subtract the base area improperly. Another error is to misinterpret the slant height as a vertical height or to use the wrong formula. Some learners also forget to multiply by 4 after computing the area of one triangular face. Using the simple formula 2 * a * l or carefully counting the triangles avoids these issues.

Final Answer:
The slant surface area of the square pyramid is 60 sq.cm.