How many sides does a regular polygon have with its interior angle as eight times its exterior angle.

Options

A. 16
B. 24
C. 18
D. 20

Answer

To solve the problem of finding how many sides a regular polygon has if its interior angle is eight times its exterior angle, we can follow these steps:

### Step 1: Understand the relationship between interior and exterior angles
For a regular polygon with \( n \) sides:
- The formula for the interior angle \( I \) is:
\[ I = \frac{(n - 2) \times 180}{n} \]
- The formula for the exterior angle \( E \) is:
\[ E = \frac{360}{n} \]

### Step 2: Set up the equation based on the problem statement
According to the problem, the interior angle is eight times the exterior angle:
\[ I = 8E \]

### Step 3: Substitute the formulas into the equation
Substituting the formulas for \( I \) and \( E \) into the equation gives:
\[ \frac{(n - 2) \times 180}{n} = 8 \times \frac{360}{n} \]

### Step 4: Simplify the equation
To eliminate \( n \) from the denominator, multiply both sides by \( n \):
\[ (n - 2) \times 180 = 8 \times 360 \]

### Step 5: Calculate the right side
Calculating the right side:
\[ 8 \times 360 = 2880 \]
So the equation now is:
\[ (n - 2) \times 180 = 2880 \]

### Step 6: Divide both sides by 180
To isolate \( n - 2 \), divide both sides by 180:
\[ n - 2 = \frac{2880}{180} \]

### Step 7: Simplify the division
Calculating the division:
\[ \frac{2880}{180} = 16 \]
So now we have:
\[ n - 2 = 16 \]

### Step 8: Solve for \( n \)
Add 2 to both sides to find \( n \):
\[ n = 16 + 2 = 18 \]

### Conclusion
Thus, the regular polygon has **18 sides**.