If the height of an equilateral triangle is 18 cm, then its area is :

Question

Accepted Answer

To determine the area of an equilateral triangle given its height, we can use the relationship between the side length and the height of an equilateral triangle.

For an equilateral triangle with side length $ s $, the height $ h $ can be expressed as:
$$ h = \frac{\sqrt{3}}{2} s $$

Given that the height $ h $ is 18 cm, we can solve for the side length $ s $:
$$ 18 = \frac{\sqrt{3}}{2} s $$
$$ s = \frac{18 \times 2}{\sqrt{3}} $$
$$ s = \frac{36}{\sqrt{3}} $$
$$ s = 12\sqrt{3} $$

Now, we calculate the area $ A $ of the equilateral triangle using the formula:
$$ A = \frac{\sqrt{3}}{4} s^2 $$

Substituting the value of $ s $:
$$ A = \frac{\sqrt{3}}{4} (12\sqrt{3})^2 $$
$$ A = \frac{\sqrt{3}}{4} \times 144 \times 3 $$
$$ A = \frac{\sqrt{3}}{4} \times 432 $$
$$ A = 108\sqrt{3} \, \text{cm}^2 $$

Thus, the area of the equilateral triangle is $ 108\sqrt{3} \, \text{cm}^2 $, which corresponds to Option 4.

**Explanation for Incorrect Options:**

1. **Option 1: $ 36\sqrt{3} \, \text{cm}^2 $** - This option is incorrect because it underestimates the area. The calculation shows that the area is significantly larger than $ 36\sqrt{3} $.

2. **Option 2: $ 72\sqrt{3} \, \text{cm}^2 $** - This option is incorrect as it also underestimates the area. The correct calculation yields a larger value.

3. **Option 3: $ 96\sqrt{3} \, \text{cm}^2 $** - This option is incorrect because it still underestimates the area. The correct area calculation results in a value of $ 108\sqrt{3} $.

Therefore, Option 4 is the correct answer, as it accurately reflects the calculated area of the equilateral triangle given the height of 18 cm.