If the length of each side of an equilateral triangle is 8 cm, what is the exact value of its area in square centimetres?

Introduction / Context:
This problem tests the standard mensuration formula for the area of an equilateral triangle. In competitive exams, equilateral triangles appear frequently because their symmetry allows neat formulas involving the square of the side and the square root of 3. Here, you are given the side length and are asked to find the exact area, not a decimal approximation, so remembering the correct formula is essential.

Given Data / Assumptions:

• The triangle is equilateral, so all three sides are equal and each interior angle is 60 degrees.
• Each side of the triangle has length 8 cm.
• The required answer is the area in square centimetres, preferably in simplest radical form involving √3.
• We assume standard Euclidean geometry and no rounding unless needed for checking.

Concept / Approach:
The area A of an equilateral triangle with side length a is given by the well known formula A = (√3 / 4) * a^2. This comes from dropping an altitude, which forms two congruent right triangles, and then applying Pythagoras theorem. Once you know this formula, you simply substitute the side length and simplify. There is no need for trigonometry or coordinate geometry here, only direct substitution and squaring of the side length.

Step-by-Step Solution:
Let the side length of the equilateral triangle be a = 8 cm.Use the formula for area: A = (√3 / 4) * a^2.Compute a^2 = 8^2 = 64.Substitute into the formula: A = (√3 / 4) * 64.Simplify 64 / 4 = 16, so A = 16√3 sq cm.

Verification / Alternative check:
To verify, you can derive the formula quickly. Drop an altitude from one vertex. This altitude splits the opposite side into two segments of length 4 cm each and creates two right triangles. The hypotenuse is 8 cm and one leg is 4 cm. Using Pythagoras theorem, height h satisfies h^2 + 4^2 = 8^2, so h^2 = 64 − 16 = 48 and h = 4√3. Area is then (1 / 2) * base * height = (1 / 2) * 8 * 4√3 = 16√3 sq cm, matching the direct formula.

Why Other Options Are Wrong:
32√3 sq cm is double the correct area and would correspond to treating 8 as the height as well as the side. The value 16 sq cm ignores √3 and is far too small for a triangle with sides of length 8 cm. The value 32 sq cm comes from incorrectly using (1 / 2) * 8 * 8. The value 64√3 sq cm is four times the correct area and arises from omitting the division by 4 in the formula.

Common Pitfalls:
A common mistake is to confuse the area formula of an equilateral triangle with that of a general triangle and write (1 / 2) * a^2 instead of (√3 / 4) * a^2. Some students also forget to square the side, using (√3 / 4) * 8 rather than (√3 / 4) * 8^2. Another error is to treat 8 as both base and height, which is not correct because the altitude is shorter than the side. Careful substitution prevents these errors.

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