The diagonal of a square is equal in length to the side of an equilateral triangle. If the area of the square is 12 cm^2, find the exact area of the equilateral triangle in square centimetres.

Introduction / Context:
This geometry question links the properties of a square and an equilateral triangle through a shared length. The diagonal of the square is given to be equal to the side of the equilateral triangle. Knowing the area of the square, you must determine the area of the equilateral triangle. This problem tests understanding of area formulas, the relationship between side and diagonal in a square, and the area formula for an equilateral triangle.

Given Data / Assumptions:

• The area of the square is 12 cm^2.
• The diagonal of the square equals the side length of an equilateral triangle.
• We need the area of the equilateral triangle in exact form.
• All lengths are in centimetres and areas in square centimetres.

Concept / Approach:
First, use the area of the square to find its side length. Then use the relation between the side and diagonal of a square: diagonal = side × √2. This diagonal is given to be the side length of the equilateral triangle. Once we know that side, we use the area formula for an equilateral triangle, which is (√3/4) × side^2. Finally, we simplify the surds to obtain the exact area in terms of √3.

Step-by-Step Solution:
Step 1: Let the side of the square be s cm.Step 2: The area of the square is s^2 and is given as 12, so s^2 = 12.Step 3: Therefore s = √12 = 2√3 cm.Step 4: The diagonal d of the square is s√2, so d = 2√3 × √2 = 2√6 cm.Step 5: This diagonal equals the side of the equilateral triangle, so the side length a of the equilateral triangle is a = 2√6 cm.Step 6: The area of an equilateral triangle is (√3/4) × a^2.Step 7: Compute a^2: a^2 = (2√6)^2 = 4 × 6 = 24.Step 8: Substitute into the area formula: Area = (√3/4) × 24 = 6√3 cm^2.

Verification / Alternative check:
As a quick check, you can compute approximate decimal values. Since √3 is about 1.732, 6√3 is about 10.392. The square has area 12 cm^2, so it is reasonable that the equilateral triangle whose side equals the diagonal of the square has slightly smaller area than the square, because its shape is different even though one of its sides is longer than the side of the square. This rough comparison supports the reasonableness of the result.

Why Other Options Are Wrong:
The value 12√3 cm^2 is double the correct area and would arise if a^2 were incorrectly taken as 48 instead of 24. The value 12√2 cm^2 belongs to a different configuration and does not follow from the equilateral triangle area formula. The value 24 cm^2 ignores the square root factor and is too large. The value 3√3 cm^2 is exactly half of the correct area. Only 6√3 cm^2 matches the correct step by step computation.

Common Pitfalls:
Learners sometimes confuse the diagonal formula and mistakenly use s^2 instead of s√2. Another error is forgetting to square both 2 and √6 when computing a^2, which leads to wrong coefficients. Misremembering the area formula for an equilateral triangle is also common. Keeping these formulas clear and substituting carefully prevents such mistakes.

Final Answer:
The area of the equilateral triangle is 6√3 cm^2.