In a regular polygon , each interior angle is thrice the exterior angle. The number of sides of the polygon is

Options

A. `6`
B. `8`
C. `10`
D. `12`

Answer

To solve the problem, we need to find the number of sides of a regular polygon where each interior angle is three times the exterior angle. Let's break it down step by step.

### Step 1: Understand the relationship between interior and exterior angles
In a regular polygon, the interior angle (I) and exterior angle (E) are related by the equation:
\[ I = 3E \]

### Step 2: Use the formulas for interior and exterior angles
The formulas for the interior and exterior angles of a regular polygon with \( n \) sides are:
- Interior angle:
\[ I = \frac{(n - 2) \times 180}{n} \]
- Exterior angle:
\[ E = \frac{360}{n} \]

### Step 3: Substitute the exterior angle into the interior angle equation
From the relationship \( I = 3E \), we can substitute the formula for the exterior angle:
\[ \frac{(n - 2) \times 180}{n} = 3 \left( \frac{360}{n} \right) \]

### Step 4: Simplify the equation
Now, we can simplify the equation:
1. Multiply both sides by \( n \) to eliminate the denominator:
\[ (n - 2) \times 180 = 3 \times 360 \]
2. Calculate \( 3 \times 360 \):
\[ (n - 2) \times 180 = 1080 \]

### Step 5: Solve for \( n \)
1. Divide both sides by 180:
\[ n - 2 = \frac{1080}{180} \]
\[ n - 2 = 6 \]
2. Add 2 to both sides:
\[ n = 6 + 2 \]
\[ n = 8 \]

### Conclusion
The number of sides of the polygon is \( n = 8 \).