If the perimeter of an equilateral triangle is 36 units, then its area is (in square units):

Correct — B. Option (b) shows 36√3.

All three sides are equal, so:
side = 36 ⁄ 3 = 12 units

Area of an equilateral triangle = (√3⁄4) × side²
= (√3⁄4) × 12² = (√3⁄4) × 144
= 36√3 square units → option (b)

Why the others are wrong

• (a)
  Option (a) shows 144√2. It keeps side² = 144 undivided and carries √2, the diagonal ratio of a square, where the equilateral constant √3⁄4 belongs.
• (c)
  Option (c) shows 36√2. The 36 is right and the surd is not: the altitude of an equilateral triangle is (√3⁄2) × side, so √3 is what survives into the area.
• (d)
  Option (d) shows 144√3, which is side² × √3 with the ÷ 4 dropped. That single omission multiplies the true area by four.

The formula is not arbitrary. Drop a perpendicular from one vertex of an equilateral triangle of side s and it bisects the base, leaving a right triangle of legs s⁄2 and h.

h² = s² − (s⁄2)² = 3s²⁄4, so h = (√3⁄2)s.

Area = ½ × base × height = ½ × s × (√3⁄2)s = (√3⁄4)s². Knowing where the √3 and the 4 come from is what stops you writing the wrong surd under pressure.

Key facts

• Area of an equilateral triangle of side s is (√3⁄4)s².
• Its altitude is (√3⁄2)s, which is the source of the √3.
• A perimeter of 36 units gives side 12 and area 36√3, about 62.4 square units.

Common traps

• Using ½ × base × side instead of ½ × base × altitude
• Writing √3 s² and forgetting the division by 4

No directly related past PYQ was found.