In plane geometry, if the length of each side of an equilateral triangle is 8 cm, what is the area of this equilateral triangle (in square centimetres)?

Introduction / Context:
This is a standard mensuration question about the area of an equilateral triangle. Such problems are common in aptitude and competitive exams and test whether you remember and can correctly apply the formula for the area of an equilateral triangle in terms of its side length.

Given Data / Assumptions:

• The triangle is equilateral (all sides and all angles equal).
• Each side has length 8 cm.
• We are asked to find the area in square centimetres.
• We assume Euclidean plane geometry and exact values with √3 left in surd form.

Concept / Approach:
For an equilateral triangle with side length a, the area formula is: Area = (√3 / 4) * a^2 This comes from drawing a height in the equilateral triangle, forming two 30°–60°–90° right triangles. One can also derive it using the general triangle area formula (1/2 * base * height) together with trigonometry for the height.

Step-by-Step Solution:
Let a = 8 cm. Use the formula Area = (√3 / 4) * a^2. Compute a^2: 8^2 = 64. Substitute: Area = (√3 / 4) * 64. Simplify 64 / 4 = 16, so Area = 16√3 sq cm.

Verification / Alternative check:
We know that an equilateral triangle with side 2 has area √3 sq units (since (√3 / 4) * 2^2 = √3). If the side is scaled up by a factor of 4 (from 2 to 8), area scales by the square of that factor, that is 4^2 = 16. Therefore the new area should be 16·√3, which matches our result of 16√3 sq cm.

Why Other Options Are Wrong:
32√3 sq cm: This is double the correct area and would correspond to a larger side length. 16 sq cm and 32 sq cm ignore the √3 factor, so they do not respect the exact geometry of an equilateral triangle.

Common Pitfalls:
Students often confuse the formula for the area of an equilateral triangle with that of a general triangle or forget the √3 factor, writing (1/2)*a^2 instead. Another mistake is squaring only part of the expression. Always remember that for equilateral triangles, √3 is always present in the exact area expression.

Final Answer:
The area of the equilateral triangle is 16√3 sq cm.