Difference between the interior and the exterior angles of regular polygon is `60^(@)`. The number of sides in the polygon is :

Options

A. 5
B. 6
C. 8
D. 9

Answer

To find the number of sides in a regular polygon where the difference between the interior and exterior angles is \(60^\circ\), we can follow these steps:

### Step 1: Define the interior and exterior angles
For a regular polygon with \(N\) sides:
- The interior angle \(I\) is given by the formula:
\[ I = \frac{(N-2) \times 180^\circ}{N} \]
- The exterior angle \(E\) is given by the formula:
\[ E = \frac{360^\circ}{N} \]

### Step 2: Set up the equation based on the difference
According to the problem, the difference between the interior and exterior angles is \(60^\circ\):
\[ I - E = 60^\circ \]

### Step 3: Substitute the formulas into the equation
Substituting the expressions for \(I\) and \(E\) into the equation gives:
\[ \frac{(N-2) \times 180^\circ}{N} - \frac{360^\circ}{N} = 60^\circ \]

### Step 4: Simplify the equation
Combine the terms on the left side:
\[ \frac{(N-2) \times 180^\circ - 360^\circ}{N} = 60^\circ \]
This simplifies to:
\[ \frac{180N - 360 - 360}{N} = 60^\circ \]
\[ \frac{180N - 720}{N} = 60^\circ \]

### Step 5: Multiply both sides by \(N\)
To eliminate the fraction, multiply both sides by \(N\):
\[ 180N - 720 = 60N \]

### Step 6: Rearrange the equation
Rearranging the equation gives:
\[ 180N - 60N = 720 \]
\[ 120N = 720 \]

### Step 7: Solve for \(N\)
Now, divide both sides by \(120\):
\[ N = \frac{720}{120} = 6 \]

### Conclusion
The number of sides in the polygon is \(N = 6\).

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