Each of the interior angles of a regular polygon is 140°. How many sides has the polygon?

Analysis

The question asks for the number of sides ($n$) of a regular polygon, given that each interior angle measures $140^\circ$.

Concept

To solve this, we use the relationship between the interior angle and the exterior angle of a polygon.

1. The sum of an interior angle and its corresponding exterior angle is always $180^\circ$ (angles on a straight line).
2. The sum of all exterior angles of any convex polygon is always $360^\circ$.
3. For a regular polygon, each exterior angle is calculated as: $$\text{Exterior Angle} = \frac{360^\circ}{n}$$

Solution

Step 1: Find the exterior angle. Since the interior angle is $140^\circ$:

Step 2: Use the exterior angle formula to find $n$. We know that:

Step 3: Solve for $n$. Multiply both sides by $n$ and divide by $40^\circ$:

The polygon has 9 sides (it is a nonagon).

Conclusion

Comparing our result with the options provided, the correct answer is A.

Exam Tip

Always remember that Exterior Angle = $360^\circ / n$. It is usually much faster to work with the exterior angle than the interior angle formula ($(n-2) \times 180 / n$) because it involves simpler division.