The sum of the measures of all the interior angles of a polygon is 2340°. How many sides does this polygon have?

Introduction / Context:
This problem checks your knowledge of the formula for the sum of interior angles of a polygon. Aptitude tests frequently include such questions to see whether you can recall the correct expression, substitute the given total angle sum, and solve for the number of sides of the polygon using simple algebra. It also reinforces understanding of how polygons generalize from triangles and quadrilaterals.

Given Data / Assumptions:

• The polygon is a simple convex polygon.
• The sum of all its interior angles is 2340°.
• We must find the number of sides n of the polygon.
• We will use the standard formula relating the number of sides and the sum of interior angles.

Concept / Approach:
For a polygon with n sides, the sum of the measures of all interior angles is given by:
Sum of interior angles = (n - 2) * 180°. This formula comes from the fact that any polygon with n sides can be divided into (n - 2) triangles, each having a sum of interior angles equal to 180°. We are given the sum and must solve for n by setting (n - 2) * 180° equal to 2340°.

Step-by-Step Solution:
Step 1: Write the formula: (n - 2) * 180° = 2340°. Step 2: Divide both sides by 180° to isolate (n - 2). Step 3: (n - 2) = 2340 / 180. Step 4: Compute 2340 / 180 = 13. Step 5: So n - 2 = 13. Step 6: Add 2 to both sides: n = 13 + 2 = 15. Step 7: Therefore, the polygon has 15 sides.

Verification / Alternative check:
To verify, plug n = 15 back into the formula. The sum of interior angles should be (15 - 2) * 180° = 13 * 180°. Calculating, 13 * 180 = 2340°, which matches the given sum exactly. This confirms our value of n and shows that there is no arithmetic error in the division or addition steps.

Why Other Options Are Wrong:
If n = 13, the sum would be (13 - 2) * 180° = 11 * 180° = 1980°, which is less than 2340°. For n = 17, the sum would be (17 - 2) * 180° = 15 * 180° = 2700°, which is greater than 2340°. For n = 11, the sum is (11 - 2) * 180° = 9 * 180° = 1620°. A hypothetical n = 19 gives an even larger sum. Only n = 15 gives exactly 2340°.

Common Pitfalls:
Students sometimes mistakenly use 180n instead of (n - 2) * 180°, which overestimates the sum. Another frequent error is confusing interior and exterior angles. For exterior angles of a convex polygon, the sum is always 360°, which is a different concept. Also, forgetting to divide by 180° properly or mishandling the arithmetic division 2340 / 180 can lead to an incorrect number of sides.

Final Answer:
Hence, the polygon has 15 sides.