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)?
Aptitude
Area
Difficulty: Easy
Choose an option
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A2 :1
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B2 : √3
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C4 :3
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D4 : √3
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E√3 : 4
Answer
Correct Answer: 4 : √3
Explanation
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:
A_s : A_t = a^2 : (√3/4 a^2).Cancel a^2 ⇒ 1 : (√3/4).Multiply both terms by 4 ⇒ 4 : √3.Verification / Alternative check:
Let a = 4 ⇒ A_s = 16; A_t = (√3/4)*16 = 4√3 ⇒ ratio = 16 : 4√3 = 4 : √3.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.