If one of the interior angles of a regular polygon is equal to `5//6` times of one of the interior angles of a regular pentagon, then the number of sides of the polygon is:

Options

A. 3
B. 4
C. 6
D. 8

Answer

To solve the problem step-by-step, we need to find the number of sides of a regular polygon given that one of its interior angles is equal to \( \frac{5}{6} \) times the interior angle of a regular pentagon.

### Step 1: Calculate the interior angle of a regular pentagon.
The formula for the interior angle \( A \) of a regular polygon with \( n \) sides is given by:
\[ A = \frac{(n-2) \times 180}{n} \]
For a pentagon, \( n = 5 \):
\[ A = \frac{(5-2) \times 180}{5} = \frac{3 \times 180}{5} = \frac{540}{5} = 108 \text{ degrees} \]

### Step 2: Find the interior angle of the regular polygon.
According to the problem, the interior angle \( A_p \) of the regular polygon is:
\[ A_p = \frac{5}{6} \times 108 \]
Calculating this gives:
\[ A_p = \frac{5 \times 108}{6} = \frac{540}{6} = 90 \text{ degrees} \]

### Step 3: Set up the equation for the interior angle of the polygon.
Let \( n \) be the number of sides of the regular polygon. We can set up the equation:
\[ \frac{(n-2) \times 180}{n} = 90 \]

### Step 4: Solve for \( n \).
Multiply both sides by \( n \):
\[ (n-2) \times 180 = 90n \]
Expanding this gives:
\[ 180n - 360 = 90n \]
Now, rearranging the equation:
\[ 180n - 90n = 360 \]
\[ 90n = 360 \]
Dividing both sides by 90:
\[ n = \frac{360}{90} = 4 \]

### Conclusion
The number of sides of the polygon is \( n = 4 \).