The height of an equilateral triangle is 18 cm. What is the area of the triangle (in square centimetres)?

Introduction / Context:
This question asks you to find the area of an equilateral triangle when its height is given. Equilateral triangles have all sides equal and all angles equal to 60°, and there is a specific relationship between the side length and the height. By using this relationship and the standard area formula, we can compute the area from the given height.

Given Data / Assumptions:

• The triangle is equilateral.
• The height h of the triangle is 18 cm.
• We must find the area in square centimetres.

Concept / Approach:
For an equilateral triangle of side length a, the height h is:
h = (√3 / 2) * a From this, we can express a in terms of h:
a = (2h) / √3 The area of an equilateral triangle is:
Area = (√3 / 4) * a^2 By substituting the expression for a into the area formula, or by first finding a numerically and then using the area formula, we can find the required area.

Step-by-Step Solution:
Step 1: Use the height relation h = (√3 / 2) * a with h = 18. Step 2: Rearrange to find a: a = (2h) / √3 = (2 * 18) / √3 = 36 / √3. Step 3: Rationalise a if desired: a = (36 * √3) / 3 = 12√3. Step 4: Now apply the area formula Area = (√3 / 4) * a^2. Step 5: Compute a^2: (12√3)^2 = 144 * 3 = 432. Step 6: Substitute into the area formula: Area = (√3 / 4) * 432. Step 7: Compute 432 / 4 = 108, giving Area = 108√3 square centimetres.

Verification / Alternative check:
Alternatively, using base b and height h, area can be written as Area = (1/2) * b * h. Here, the base b is equal to the side length a = 12√3. Then:
Area = (1/2) * 12√3 * 18 = 6√3 * 18 = 108√3 sq cm. This matches the area obtained from the equilateral triangle formula, so the result is confirmed.

Why Other Options Are Wrong:

• 36√3 sq m and 96√3 sq m: These are in square metres instead of square centimetres and also have incorrect numerical values.
• 108 sq cm: This omits the √3 factor and therefore underestimates the area.
• 54√3 sq cm: This is exactly half of the correct area and may result from improperly using base or height.

Common Pitfalls:
Students often confuse side length and height in equilateral triangles, or they forget the exact relationships involving √3. Another common error is mixing units or miscomputing squares of expressions with √3. Carefully using the formulas h = (√3 / 2) * a and Area = (√3 / 4) * a^2, along with neat algebra, helps avoid these mistakes.

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