The ratio of an interior angle to an exterior angle of a regular polygon is `5 : 2`. What is the number of sides of the polygon?

Answer

To find the number of sides of a regular polygon given that the ratio of its interior angle to its exterior angle is 5:2, we can follow these steps:

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

### Step 2: Set up the ratio
According to the problem, the ratio of the interior angle to the exterior angle is given as \( 5:2 \). We can express this as:
\[ \frac{I}{E} = \frac{5}{2} \]

### Step 3: Substitute the formulas into the ratio
Substituting the formulas for \( I \) and \( E \) into the ratio gives:
\[ \frac{\frac{(n - 2) \times 180}{n}}{\frac{360}{n}} = \frac{5}{2} \]

### Step 4: Simplify the ratio
When we simplify the left side, we can multiply both sides by \( n \):
\[ \frac{(n - 2) \times 180}{360} = \frac{5}{2} \]
This simplifies to:
\[ \frac{(n - 2)}{2} = \frac{5}{2} \]

### Step 5: Cross-multiply to solve for \( n \)
Cross-multiplying gives:
\[ 2(n - 2) = 5 \]
Expanding this results in:
\[ 2n - 4 = 5 \]

### Step 6: Solve for \( n \)
Adding 4 to both sides:
\[ 2n = 9 \]
Now, divide by 2:
\[ n = \frac{9}{2} = 4.5 \]
This is incorrect; let's recheck the steps.

### Step 7: Correct the equation
Going back to the simplified ratio:
\[ (n - 2) \times 180 = 5 \times 360 / 2 \]
This simplifies to:
\[ (n - 2) \times 180 = 900 \]
Now divide both sides by 180:
\[ n - 2 = 5 \]
Adding 2 to both sides gives:
\[ n = 7 \]

### Conclusion
The number of sides of the polygon is \( n = 7 \).