The exterior angle of a regular polygon is one-third of its interior angle. How many sides has the polygon?

Answer

To find the number of sides of a regular polygon where the exterior angle is one-third of its interior angle, we can follow these steps:

### Step 1: Understand the relationship between interior and exterior angles
The exterior angle (E) of a polygon is related to its interior angle (I) by the equation:
\[ E = 180 - I \]
According to the problem, we know that:
\[ E = \frac{1}{3} I \]

### Step 2: Set up the equation
From the relationship above, we can substitute \( E \) in the equation:
\[ \frac{1}{3} I = 180 - I \]

### Step 3: Solve for I
To eliminate the fraction, multiply the entire equation by 3:
\[ I = 540 - 3I \]

Now, add \( 3I \) to both sides:
\[ I + 3I = 540 \]
\[ 4I = 540 \]

Now, divide both sides by 4:
\[ I = \frac{540}{4} = 135 \]

### Step 4: Find the exterior angle
Now that we have the interior angle, we can find the exterior angle using:
\[ E = 180 - I \]
\[ E = 180 - 135 = 45 \]

### Step 5: Use the exterior angle to find the number of sides
The formula for the exterior angle of a regular polygon is:
\[ E = \frac{360}{n} \]
where \( n \) is the number of sides. We can set up the equation:
\[ 45 = \frac{360}{n} \]

### Step 6: Solve for n
To find \( n \), rearrange the equation:
\[ n = \frac{360}{45} \]
Calculating this gives:
\[ n = 8 \]

### Conclusion
Thus, the polygon has **8 sides**.

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