What is the measure, in degrees, of each exterior angle of a regular polygon that has 9 sides?

Introduction / Context:
This question concerns the exterior angles of a regular polygon. A regular polygon has all sides and all interior angles equal. The sum of all exterior angles of any convex polygon is always 360 degrees. Therefore, for a regular polygon, each exterior angle can be found by dividing 360 by the number of sides. Here, the polygon has 9 sides, so we apply this standard rule.

Given Data / Assumptions:

• Polygon is regular with n = 9 sides.
• All exterior angles of a regular polygon are equal.
• Sum of all exterior angles of any convex polygon = 360 degrees.

Concept / Approach:
The key concept is that the sum of the exterior angles, taken one at each vertex, is 360 degrees irrespective of the number of sides. For a regular polygon, each exterior angle is equal, so each is simply 360 / n. Substituting n = 9 gives the measure of each exterior angle. No advanced geometry is needed, just a clear understanding of this property.

Step-by-Step Solution:
Number of sides n = 9.Sum of all exterior angles = 360 degrees.Each exterior angle in a regular polygon = 360 / n.So each exterior angle = 360 / 9.Compute 360 / 9 = 40 degrees.Therefore, each exterior angle of a regular 9 sided polygon is 40 degrees.

Verification / Alternative check:
We can also relate interior and exterior angles. For any regular polygon, interior angle + exterior angle at a vertex = 180 degrees. If each exterior angle is 40 degrees, then each interior angle would be 180 - 40 = 140 degrees. Using the interior angle formula, interior angle = ((n - 2) * 180) / n = (7 * 180) / 9 = 140 degrees, which matches. This confirms that 40 degrees is correct for the exterior angle.

Why Other Options Are Wrong:
36 degrees would be correct for a regular polygon with 10 sides, not 9. 30 degrees corresponds to 12 sides, 24 degrees to 15 sides, and 60 degrees to 6 sides. None of these match n = 9. Only 40 degrees gives a sum of 9 * 40 = 360 degrees, which satisfies the basic property of exterior angles.

Common Pitfalls:
A common mistake is to use formulas meant for interior angles when the question asks about exterior angles. Some learners also incorrectly add 360 and divide by (n - 2). Remember that the sum of exterior angles is always 360 degrees, independent of n. Keeping this rule in mind and directly dividing by the number of sides ensures a quick and accurate solution.

Final Answer:
40