Find the area (in cm 2 ) of an equilateral triangle whose sides are 18 cm.

Correct — D. The area of an equilateral triangle of side s is (√3⁄4) × s². The four options are printed as images, all of the form (number)√3.

s = 18 cm, so s² = 18 × 18 = 324
324 ÷ 4 = 81
Area = 81√3 cm² → option (d).

Cross-check with base and height: the altitude is (√3⁄2) × 18 = 9√3, and ½ × 18 × 9√3 = 81√3. The same figure.

Why the others are wrong

• (a)
  The coefficient can only be s² ÷ 4 = 324 ÷ 4 = 81. 77√3 would need a side of √308 ≈ 17.55 cm, not the 18 cm the stem gives.
• (b)
  75√3 is the area of an equilateral triangle of side √300 = 10√3 ≈ 17.32 cm. The side here is 18 cm.
• (c)
  79√3 would need s² = 316, so a side near 17.78 cm. Squaring 18 gives 324, and 324 ÷ 4 is 81.

For an equilateral triangle of side s the area is (√3⁄4)s², and it falls out of the ordinary base-and-height formula.

The altitude drops to the midpoint of the base, splitting the triangle into two 30-60-90 triangles whose long leg is (√3⁄2)s. Then ½ × s × (√3⁄2)s = (√3⁄4)s².

The √3 never leaves the answer, which is why all four options here are images of surds. An answer with no √3 in it comes from the wrong formula.

Key facts

• The area of an equilateral triangle of side s is (√3⁄4)s².
• Its altitude is (√3⁄2)s, so a side of 18 cm gives an altitude of 9√3 cm.
• For s = 18 cm the area is 81√3 cm², which is about 140.3 cm².

Common traps

• Treating the 18 cm side as the height and computing ½ × 18 × 18 = 162.
• Dividing by 2 rather than 4 and reporting 162√3.
• Replacing √3 by 1.73 and hunting for a decimal option when every option is a surd.

No directly related past PYQ was found.