If difference between exterior and interior angles of a polygon is `60^@`, then find the number of sides in the polygon.

Options

A. 4
B. 5
C. 6
D. 7

Answer

To solve the problem of finding the number of sides in a polygon given that the difference between its exterior and interior angles is \(60^\circ\), we can follow these steps:

### Step 1: Understand the relationship between interior and exterior angles
The interior angle \(I\) of a polygon with \(n\) sides can be calculated using the formula:
\[ I = \frac{(n-2) \times 180^\circ}{n} \]
The exterior angle \(E\) of a polygon can be calculated using the formula:
\[ E = \frac{360^\circ}{n} \]

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

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

### Step 4: Simplify the equation
To simplify, we can combine the terms on the left side:
\[ \frac{360^\circ - (n-2) \times 180^\circ}{n} = 60^\circ \]
This simplifies to:
\[ \frac{360^\circ - 180^\circ n + 360^\circ}{n} = 60^\circ \]
\[ \frac{720^\circ - 180^\circ n}{n} = 60^\circ \]

### Step 5: Eliminate the fraction by multiplying both sides by \(n\)
Multiplying both sides by \(n\) gives:
\[ 720^\circ - 180^\circ n = 60^\circ n \]

### Step 6: Rearrange the equation
Rearranging the equation to isolate \(n\):
\[ 720^\circ = 180^\circ n + 60^\circ n \]
\[ 720^\circ = 240^\circ n \]

### Step 7: Solve for \(n\)
Now, divide both sides by \(240^\circ\):
\[ n = \frac{720^\circ}{240^\circ} = 3 \]

### Step 8: Conclusion
Thus, the number of sides in the polygon is:
\[ n = 6 \]