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’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
Since it is an equilateral triangle, all three sides are equal. Let:
- \( A = 10 \, \text{cm} \)
- \( B = 10 \, \text{cm} \)
- \( C = 10 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{A + B + C}{2} \]
Substituting the values:
\[ s = \frac{10 + 10 + 10}{2} = \frac{30}{2} = 15 \, \text{cm} \]

### Step 3: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s - A)(s - B)(s - C)} \]
Substituting the values of \( s \), \( A \), \( B \), and \( C \):
\[ A = \sqrt{15(15 - 10)(15 - 10)(15 - 10)} \]
This simplifies to:
\[ A = \sqrt{15 \times 5 \times 5 \times 5} \]

### Step 4: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{15 \times 125} = \sqrt{1875} \]
Now, we can simplify \( 1875 \):
\[ 1875 = 25 \times 75 = 25 \times 25 \times 3 = 625 \times 3 \]
Thus,
\[ A = \sqrt{625 \times 3} = 25\sqrt{3} \]

### Step 5: State the final area
The area of the equilateral triangle is:
\[ A = 25\sqrt{3} \, \text{cm}^2 \]

### Summary
The area of the equilateral triangle with each side measuring 10 cm is \( 25\sqrt{3} \, \text{cm}^2 \).

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