What is the sum of all interior angles (in degrees) of a polygon that has 14 sides?

Introduction / Context:
This problem involves a fundamental result from polygon geometry. It asks for the sum of all interior angles of a polygon with 14 sides, which is also called a 14-gon. The relationship between the number of sides of a polygon and the sum of its interior angles is standard and is widely used in school mathematics and aptitude tests.

Given Data / Assumptions:
- The polygon has n = 14 sides.- It is a simple polygon (no self intersections).- We are asked to find the sum of its interior angles, not the measure of each individual angle.- We use the standard formula for the sum of interior angles of an n sided polygon.

Concept / Approach:
The formula for the sum of all interior angles of a polygon with n sides is: Sum = (n - 2) * 180 degrees. This formula can be derived by dividing the polygon into (n - 2) triangles, each contributing 180 degrees. Once n is known, we simply substitute and compute the product. This is a straightforward application of the formula and does not require advanced geometry.

Step-by-Step Solution:
Step 1: Identify the number of sides of the polygon: n = 14.Step 2: Recall the formula for sum of interior angles: Sum = (n - 2) * 180 degrees.Step 3: Substitute n = 14 into the formula.Step 4: Compute (14 - 2) = 12.Step 5: Now compute Sum = 12 * 180 degrees.Step 6: Calculate 12 * 180 = 2160 degrees.Step 7: Therefore, the sum of all interior angles of a 14 sided polygon is 2160 degrees.

Verification / Alternative check:
To verify, consider smaller polygons. For a triangle (n = 3), the formula gives (3 - 2) * 180 = 180 degrees, which is correct. For a quadrilateral (n = 4), it gives (4 - 2) * 180 = 360 degrees, also correct. Extending this pattern to n = 14 is consistent and gives 2160 degrees, so the formula is trusted. There is no contradiction with the known values for simpler polygons, which supports the calculation.

Why Other Options Are Wrong:
- 2520°: This corresponds to (n - 2) * 180 = 2520, which would give n - 2 = 14 and n = 16, not 14.- 2880°: This would mean (n - 2) * 180 = 2880, giving n = 18, not 14.- 3240°: This corresponds to a polygon with even more sides, not a 14-gon.- 1980°: This does not correspond to any integer n in the formula (n - 2) * 180.

Common Pitfalls:
One common mistake is to apply the formula for the measure of each interior angle of a regular polygon instead of the sum of all angles. Another error is to use n * 180 instead of (n - 2) * 180. Also, some might miscalculate (14 - 2) or 12 * 180 under time pressure. Being careful with substitution and arithmetic avoids these issues.

Final Answer:
The sum of all interior angles of a 14 sided polygon is 2160°.