What is the area (in sq. cm) of an equilateral triangle whose side is 12 cm?

Options

A. 24√3
B. 36√3
C. 42√3
D. 48√3

Answer

To find the area of an equilateral triangle with a side length of 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 the altitude (height) bisects the base.

### Step 2: Draw the triangle and label it
Let the equilateral triangle be ABC, where AB = AC = BC = 12 cm. Draw the altitude AD from vertex A to base BC, which meets BC at point D.

### Step 3: Find the length of BD and DC
Since AD is the altitude and bisects BC, we have:
- BD = DC = BC / 2 = 12 cm / 2 = 6 cm.

### Step 4: Apply the Pythagorean theorem
In triangle ABD, we can apply the Pythagorean theorem:
- AB² = AD² + BD²
- (12 cm)² = AD² + (6 cm)²

### Step 5: Solve for AD (the height)
Substituting the values:
- 144 = AD² + 36
- AD² = 144 - 36
- AD² = 108
- AD = √108 = √(36 * 3) = 6√3 cm.

### Step 6: Calculate the area of the triangle
The area (A) of triangle ABC can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base BC = 12 cm and the height AD = 6√3 cm.
- A = (1/2) × 12 × 6√3
- A = 6 × 6√3
- A = 36√3 cm².

### Final Answer
The area of the equilateral triangle is **36√3 cm²**.

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