Calculate the area of an equilateral triangle, whose height is 20 cm.

Answer

To calculate the area of an equilateral triangle given its height, we can follow these steps:

### Step 1: Understand the formula for the area of a triangle
The area \( A \) of any triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

### Step 2: Identify the given height
In this problem, we are given that the height \( h \) of the equilateral triangle is 20 cm.

### Step 3: Relate the height to the side of the triangle
For an equilateral triangle, the relationship between the height \( h \) and the side \( s \) is given by:
\[ h = \frac{\sqrt{3}}{2} s \]
From this, we can express the side \( s \) in terms of the height \( h \):
\[ s = \frac{2h}{\sqrt{3}} \]

### Step 4: Substitute the height into the formula for the side
Substituting \( h = 20 \) cm into the equation:
\[ s = \frac{2 \times 20}{\sqrt{3}} = \frac{40}{\sqrt{3}} \text{ cm} \]

### Step 5: Calculate the area using the side length
Now, we can use the formula for the area of an equilateral triangle:
\[ A = \frac{\sqrt{3}}{4} s^2 \]
Substituting \( s = \frac{40}{\sqrt{3}} \):
\[ A = \frac{\sqrt{3}}{4} \left(\frac{40}{\sqrt{3}}\right)^2 \]

### Step 6: Simplify the area calculation
Calculating \( s^2 \):
\[ s^2 = \left(\frac{40}{\sqrt{3}}\right)^2 = \frac{1600}{3} \]
Now substituting back into the area formula:
\[ A = \frac{\sqrt{3}}{4} \times \frac{1600}{3} = \frac{1600\sqrt{3}}{12} = \frac{400\sqrt{3}}{3} \]

### Step 7: Calculate the numerical value of the area
Using \( \sqrt{3} \approx 1.732 \):
\[ A \approx \frac{400 \times 1.732}{3} \approx \frac{692.8}{3} \approx 230.93 \text{ cm}^2 \]

### Final Answer
Thus, the area of the equilateral triangle is approximately:
\[ \boxed{230.94 \text{ cm}^2} \]

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