What is the area (in square centimetres) of an equilateral triangle whose side length is 14 cm?

Introduction / Context:
Equilateral triangles are triangles with all sides equal and all interior angles equal to 60°. There is a standard formula for the area of an equilateral triangle in terms of its side length. This formula is frequently used in aptitude and competitive exams because it allows quick calculation without needing to derive the height separately each time.

Given Data / Assumptions:

• The triangle is equilateral.
• Each side of the triangle is 14 cm long.
• We need to calculate the area in square centimetres.

Concept / Approach:
The standard area formula for an equilateral triangle with side length a is:
Area = (√3 / 4) * a^2 This formula is derived from drawing the altitude, which splits the equilateral triangle into two right triangles, and then using the Pythagorean theorem to express the height in terms of the side length. Once we know the formula, we can substitute the given side length directly.

Step-by-Step Solution:
Step 1: Identify the side length a = 14 cm. Step 2: Use the formula Area = (√3 / 4) * a^2. Step 3: Compute a^2: 14^2 = 196. Step 4: Substitute into the formula: Area = (√3 / 4) * 196. Step 5: Simplify 196 / 4 = 49. Step 6: Therefore, Area = 49√3 square centimetres.

Verification / Alternative check:
We can verify by computing the height first. In an equilateral triangle of side a, the height h is given by h = (√3 / 2) * a. For a = 14, h = (√3 / 2) * 14 = 7√3. The area is then (1/2) * base * height = (1/2) * 14 * 7√3 = 7 * 7√3 = 49√3. This matches the formula-based result and confirms the area.

Why Other Options Are Wrong:

• 98√3: This is double the correct area, possibly from forgetting to divide by 4 in the formula.
• (49/2)√3 and (49/4)√3: These values underestimate the area, coming from misuse of the denominator factor.
• 196: This is simply a^2 and ignores the √3/4 factor entirely.

Common Pitfalls:
Some students mistakenly treat an equilateral triangle like a right triangle and miscalculate the height. Others forget the exact formula and either miss the √3 factor or the 1/4 factor. Memorising the compact formula Area = (√3 / 4) * a^2 is very helpful and avoids repeated derivations during timed exams.

Final Answer:
Thus, the area of the equilateral triangle is 49√3 sq cm.