Find the area and the height of an equilateral triangle whose each side measures :
10 m

Answer

To find the area and height of an equilateral triangle with each side measuring 10 meters, we can follow these steps:

### Step 1: Understand the properties of an equilateral triangle
An equilateral triangle has all three sides equal and all angles equal to 60 degrees.

### Step 2: Use the formula for the area of an equilateral triangle
The formula for the area \( A \) of an equilateral triangle is given by:
\[ A = \frac{\sqrt{3}}{4} \times \text{side}^2 \]
Here, the side length is 10 m.

### Step 3: Substitute the side length into the area formula
Substituting the side length into the formula:
\[ A = \frac{\sqrt{3}}{4} \times 10^2 \]
\[ A = \frac{\sqrt{3}}{4} \times 100 \]

### Step 4: Simplify the area calculation
Calculating further:
\[ A = \frac{100\sqrt{3}}{4} \]
\[ A = 25\sqrt{3} \text{ m}^2 \]

### Step 5: Find the height of the triangle
The height \( h \) of an equilateral triangle can also be calculated using the area formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
For our triangle, the base is 10 m. We already calculated the area as \( 25\sqrt{3} \text{ m}^2 \).

### Step 6: Set up the equation for height
Substituting the known values into the area formula:
\[ 25\sqrt{3} = \frac{1}{2} \times 10 \times h \]

### Step 7: Solve for height
Rearranging the equation gives:
\[ 25\sqrt{3} = 5h \]
Dividing both sides by 5:
\[ h = \frac{25\sqrt{3}}{5} \]
\[ h = 5\sqrt{3} \text{ m} \]

### Conclusion
Thus, the area of the equilateral triangle is \( 25\sqrt{3} \text{ m}^2 \) and the height is \( 5\sqrt{3} \text{ m} \).

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