The height of an equilateral triangle is 18 cm. Using properties of equilateral triangles and the relationship between side and height, what is its area (in cm²)?

Introduction:
Here we are given the height of an equilateral triangle, not its side, and asked to find the area. This uses the geometric relationship between the side length and the height of an equilateral triangle, combined with the basic triangle area formula.

Given Data / Assumptions:

• The triangle is equilateral.
• Height h = 18 cm.
• We must find its area in square centimetres.

Concept / Approach:
For an equilateral triangle of side a, the height h is related by: h = (√3 / 2) * a. We can solve this for a in terms of h, then compute area using either: A = (1/2) * base * height, or A = (√3 / 4) * a². Both methods give the same answer when used correctly.

Step-by-Step Solution:
Given h = 18 cm and h = (√3 / 2) * a. Solve for a: a = (2h) / √3 = (2 * 18) / √3 = 36 / √3. Rationalize: 36 / √3 = (36√3) / 3 = 12√3 cm. So side a = 12√3 cm. Now use area formula A = (1/2) * base * height. Base = a = 12√3, height = 18. So A = (1/2) * 12√3 * 18 = 6√3 * 18. Compute 6 * 18 = 108, so A = 108√3 cm².

Verification / Alternative check:
Alternatively, use A = (√3 / 4) * a² with a = 12√3. Then a² = (12√3)² = 144 * 3 = 432. So A = (√3 / 4) * 432 = (432√3) / 4 = 108√3 cm². This matches the previous result.

Why Other Options Are Wrong:
Options with square metres are incorrect units. The value 108 sq. cm. ignores the √3 factor, which is essential due to the geometry of equilateral triangles. Only 108√3 sq. cm. fits both the correct formula and the given height.

Common Pitfalls:
Errors often include forgetting the √3 in the height formula, mixing units, or failing to square the side correctly. Another mistake is using the height directly as a side in the equilateral triangle area formula. Carefully distinguishing between side and height avoids confusion.

Final Answer:
The area of the equilateral triangle is 108√3 sq. cm.