Each side of an equilateral triangle measure 10 cm. Find the area of the triangle .

Answer

To find the area of an equilateral triangle with each side measuring 10 cm, we can use Heron's formula. Here are the steps to solve the problem:

### Step 1: Identify the sides of the triangle
Since it is an equilateral triangle, all sides are equal. Let the length of each side be \( a = 10 \) cm.

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of a triangle is given by the formula:
\[ s = \frac{a + b + c}{2} \]
For an equilateral triangle, \( a = b = c = 10 \) cm. Thus,
\[ s = \frac{10 + 10 + 10}{2} = \frac{30}{2} = 15 \text{ cm} \]

### Step 3: Apply Heron's formula
Heron's formula for the area \( A \) of a triangle is:
\[ A = \sqrt{s(s - a)(s - b)(s - c)} \]
Since \( a = b = c = 10 \) cm, we can substitute:
\[ A = \sqrt{s(s - a)(s - a)(s - a)} = \sqrt{s(s - a)^3} \]
Substituting the values we have:
\[ A = \sqrt{15(15 - 10)(15 - 10)(15 - 10)} = \sqrt{15 \times 5 \times 5 \times 5} \]

### Step 4: Simplify the expression
Now, calculate:
\[ A = \sqrt{15 \times 125} = \sqrt{1875} \]
We can simplify \( \sqrt{1875} \):
\[ 1875 = 25 \times 75 = 25 \times 25 \times 3 = 625 \times 3 \]
Thus,
\[ A = 25 \sqrt{3} \]

### Step 5: Calculate the numerical value
Using \( \sqrt{3} \approx 1.732 \):
\[ A \approx 25 \times 1.732 \approx 43.3 \text{ cm}^2 \]

### Final Answer
The area of the equilateral triangle is approximately \( 43.3 \text{ cm}^2 \).

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