Q.22

The correct area of an equilateral triangle with side length
a is
(√3/4)a², so option (A) is correct.

Question statement

The question asks: “The area of an equilateral triangle with sides of length a is ?”

• (A) (√3/4)a²
• (B) (√3/2)a²
• (C) (1/2)a²
• (D) (1/√2)a²

For an equilateral triangle (all sides equal, all angles 60°), the standard area formula is
(√3/4)a².

Derivation of the area formula

1. In an equilateral triangle of side a, draw an altitude from one vertex to the opposite side. This altitude splits the triangle into two right triangles with hypotenuse a and base a/2.
2. By Pythagoras, the height h satisfies
   h² = a² − (a/2)² = a² − a²/4 = 3a²/4, so h = (√3/2)a.
3. Area of a triangle is (1/2) × base × height.
4. Here base = a and height = (√3/2)a, so
   Area = (1/2) × a × (√3/2)a = (√3/4)a².

Thus, the formula for the area of an equilateral triangle with side a is (√3/4)a², matching option (A).

Explanation of each MCQ option

Option (A): (√3/4)a²

This option exactly matches the derived formula using base–height and other standard derivations such as trigonometry and Heron’s formula, so it is the correct choice.

Textbooks and formula sheets list the area of an equilateral triangle of side a as (√3/4)a², confirming that option (A) is correct.

Option (B): (√3/2)a²

The expression (√3/2)a² represents a × h where h = (√3/2)a, but the area formula still requires the extra factor 1/2.

Numerically, this value is twice the correct area, so option (B) overestimates the area and is incorrect.

Option (C): (1/2)a²

This option looks like someone used the formula (1/2) × base × height but mistakenly took the height equal to the side a instead of (√3/2)a.

Because it ignores the √3/2 factor, this option underestimates the area of the equilateral triangle and is therefore wrong.

Option (D): (1/√2)a²

The coefficient 1/√2 does not appear in standard formulas for equilateral triangles, where the characteristic constants are √3/2 for the height and √3/4 for the area.

Since this value does not come from Pythagoras, trigonometry, or Heron’s formula for an equilateral triangle, option (D) is also incorrect.

SEO‑friendly introduction

For students searching “area of an equilateral triangle with sides of length a,” understanding the right formula and its derivation is essential for solving geometry MCQs accurately.

This article explains the formula (√3/4)a² in a clear, step‑by‑step way and shows why only option (A) gives the correct area when the equilateral triangle has side length a.