If each interior angle is double of each exterior angle of a regular polygon with n sides, then the value of n is

Options

A. 8
B. 10
C. 5
D. 6

Answer

To solve the problem, we need to find the number of sides \( n \) of a regular polygon given that each interior angle is double that of each exterior angle. Let's go through the solution step by step.

### Step 1: Define the Angles
Let:
- Each interior angle be \( I \)
- Each exterior angle be \( E \)

According to the problem, we have the relationship:
\[ I = 2E \]

### Step 2: Relationship Between Interior and Exterior Angles
We know that the sum of an interior angle and its corresponding exterior angle is:
\[ I + E = 180^\circ \]

### Step 3: Substitute the First Equation into the Second
Substituting \( I = 2E \) into the equation \( I + E = 180^\circ \):
\[ 2E + E = 180^\circ \]
This simplifies to:
\[ 3E = 180^\circ \]

### Step 4: Solve for the Exterior Angle
Now, divide both sides by 3 to find \( E \):
\[ E = \frac{180^\circ}{3} = 60^\circ \]

### Step 5: Use the Exterior Angle to Find the Number of Sides
The sum of all exterior angles of any polygon is \( 360^\circ \). The formula to find the number of sides \( n \) of a polygon based on its exterior angle \( E \) is:
\[ n = \frac{360^\circ}{E} \]
Substituting \( E = 60^\circ \):
\[ n = \frac{360^\circ}{60^\circ} = 6 \]

### Conclusion
Thus, the value of \( n \) is \( 6 \), meaning the polygon is a hexagon.