In a regular polygon, the exterior and interior angles are in the ra tio 1:4. The number of sides of the polygon is

Options

A. 10
B. 12
C. 15
D. 16

Answer

To find the number of sides of a regular polygon where the exterior and interior angles are in the ratio of 1:4, we can follow these steps:

### Step 1: Define the exterior and interior angles
The exterior angle of a regular polygon can be calculated using the formula:
\[ \text{Exterior Angle} = \frac{360}{n} \]
where \( n \) is the number of sides of the polygon.

The interior angle can be calculated using the formula:
\[ \text{Interior Angle} = \frac{(n - 2) \times 180}{n} \]

### Step 2: Set up the ratio
According to the problem, the ratio of the exterior angle to the interior angle is given as 1:4. This can be expressed mathematically as:
\[ \frac{\text{Exterior Angle}}{\text{Interior Angle}} = \frac{1}{4} \]
Substituting the formulas for the exterior and interior angles, we have:
\[ \frac{\frac{360}{n}}{\frac{(n - 2) \times 180}{n}} = \frac{1}{4} \]

### Step 3: Simplify the equation
This simplifies to:
\[ \frac{360}{(n - 2) \times 180} = \frac{1}{4} \]
Cross-multiplying gives:
\[ 360 \times 4 = (n - 2) \times 180 \]
This simplifies to:
\[ 1440 = (n - 2) \times 180 \]

### Step 4: Solve for \( n \)
Now, divide both sides by 180:
\[ \frac{1440}{180} = n - 2 \]
Calculating the left side:
\[ 8 = n - 2 \]
Adding 2 to both sides gives:
\[ n = 10 \]

### Conclusion
The number of sides of the polygon is \( n = 10 \).

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