Find the area and the height of an equilateral triangle whose each side measures :
12 cm

Answer

To find the area and height of an equilateral triangle with each side measuring 12 cm, we can follow these steps:

### Step 1: Understand the properties of an equilateral triangle
An equilateral triangle has all three sides equal and all three angles equal to 60 degrees. In this case, each side measures 12 cm.

### Step 2: Calculate the area of the equilateral triangle
The formula for the area \( A \) of an equilateral triangle with side length \( a \) is given by:

\[ A = \frac{\sqrt{3}}{4} \times a^2 \]

Substituting \( a = 12 \) cm:

\[ A = \frac{\sqrt{3}}{4} \times (12)^2 \]

Calculating \( 12^2 \):

\[ 12^2 = 144 \]

Now substitute back into the area formula:

\[ A = \frac{\sqrt{3}}{4} \times 144 \]

Dividing \( 144 \) by \( 4 \):

\[ A = \sqrt{3} \times 36 \]

Thus, the area is:

\[ A = 36\sqrt{3} \text{ cm}^2 \]

### Step 3: Calculate the height of the equilateral triangle
The height \( h \) of an equilateral triangle can also be calculated using the area. The area can also be expressed in terms of the base and height:

\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]

Here, the base is one side of the triangle, which is 12 cm. Let \( h \) be the height:

\[ 36\sqrt{3} = \frac{1}{2} \times 12 \times h \]

Simplifying the right side:

\[ 36\sqrt{3} = 6h \]

Now, solve for \( h \):

\[ h = \frac{36\sqrt{3}}{6} \]

Calculating:

\[ h = 6\sqrt{3} \text{ cm} \]

### Final Results
- The area of the equilateral triangle is \( 36\sqrt{3} \text{ cm}^2 \).
- The height of the equilateral triangle is \( 6\sqrt{3} \text{ cm} \).

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