Compare Areas — Square of side a vs Equilateral Triangle of side a: What is the ratio of the area of a square (side a) to the area of an equilateral triangle (side a)?

Introduction / Context:This comparison leverages two standard area formulas at the same scale (side length a). It checks familiarity with special-triangle areas and the ability to express a clean ratio without units, since the common factor a^2 cancels out.

Given Data / Assumptions:

• Square of side a ⇒ Area A_s = a^2
• Equilateral triangle of side a ⇒ Area A_t = (√3/4) * a^2
• We seek A_s : A_t

Concept / Approach:Form the ratio and cancel a^2. Express the result in simplest radical form. Since both shapes share the same side length, unit-free comparison is straightforward and does not require numeric substitution unless verifying.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 2:1 and 4:3 are integer ratios unrelated to √3/4.
• 2:√3 halves the correct first term.
• √3:4 inverts the desired ratio.

Common Pitfalls:

• Using the triangle area as (1/2)ab without recognizing 60°; the special equilateral result is (√3/4)a^2.

Final Answer:4 : √3.