Find the number of sides of a regular polygon whose each exterior angle has a measure of 45°

Question

Accepted Answer

To determine the number of sides of a regular polygon where each exterior angle measures 45°, we can use the formula for the exterior angle of a regular polygon:

$$ \text{Exterior Angle} = \frac{360°}{n} $$

where $ n $ is the number of sides of the polygon.

Given that each exterior angle is 45°, we can set up the equation:

$$ 45° = \frac{360°}{n} $$

Solving for $ n $, we multiply both sides by $ n $ to get:

$$ 45° \times n = 360° $$

Then, divide both sides by 45°:

$$ n = \frac{360°}{45°} $$

$$ n = 8 $$

Therefore, the polygon has 8 sides, which corresponds to Option 1.

**Explanation for Other Options:**

- **Option 2 (10 sides):** If the polygon had 10 sides, the measure of each exterior angle would be:

$$ \frac{360°}{10} = 36° $$

This does not match the given exterior angle of 45°, so Option 2 is incorrect.

- **Option 3 (6 sides):** If the polygon had 6 sides, the measure of each exterior angle would be:

$$ \frac{360°}{6} = 60° $$

This does not match the given exterior angle of 45°, so Option 3 is incorrect.

- **Option 4 (4 sides):** If the polygon had 4 sides, the measure of each exterior angle would be:

$$ \frac{360°}{4} = 90° $$

This does not match the given exterior angle of 45°, so Option 4 is incorrect.

In conclusion, the only option that satisfies the condition of each exterior angle being 45° is Option 1, which is a polygon with 8 sides.