What is the measure of each interior angle of a regular dodecagon (a regular polygon with 12 sides)?

Introduction / Context:
This question is about regular polygons and interior angles. A regular dodecagon is a 12 sided polygon with all sides and all interior angles equal. The problem tests your ability to recall or derive the general formula for the interior angle of a regular n sided polygon and apply it for n = 12.

Given Data / Assumptions:

• The polygon is regular and has 12 sides (a dodecagon).
• All interior angles are equal.
• We need the measure of one interior angle.

Concept / Approach:
The sum of all interior angles of an n sided polygon is (n − 2) * 180°. For a regular polygon, each interior angle is this sum divided by n. Therefore, the measure of each interior angle of a regular n sided polygon is ((n − 2) * 180°) / n. Here, n = 12, so we substitute to find the measure of each interior angle of the regular dodecagon.

Step-by-Step Solution:
Step 1: Write the formula for the sum of interior angles of an n sided polygon: Sum = (n − 2) * 180°. Step 2: For a dodecagon, n = 12. Sum of interior angles = (12 − 2) * 180° = 10 * 180°. Step 3: Compute 10 * 180° = 1800°. Step 4: Because the dodecagon is regular, all 12 interior angles are equal. Step 5: Measure of each interior angle = total sum / number of angles = 1800° / 12. Step 6: Divide: 1800° / 12 = 150°.

Verification / Alternative check:
You can cross check with the formula for a regular polygon directly: interior angle = ((n − 2) * 180°) / n. Substituting n = 12 gives ((12 − 2) * 180°) / 12 = (10 * 180°) / 12 = 1800° / 12 = 150°, matching the previous calculation. This confirms the result is consistent.

Why Other Options Are Wrong:
An angle of 120° corresponds to a regular hexagon (n = 6), not a dodecagon. An angle of 140° would correspond to n = 9 (a non integer calculation shows it does not match 12 sides), and 144° is the interior angle of a regular decagon (10 sided polygon), not of a 12 sided polygon. Only 150° matches the correct formula for n = 12.

Common Pitfalls:
Students sometimes confuse the formula for the sum of interior angles with the measure of one interior angle, or they mistakenly use 180° / n instead of ((n − 2) * 180°) / n. Another error is to think that the interior angle of a regular polygon is always 180° − 360° / n without carrying out the subtraction carefully. Remember that the exterior angle of a regular n sided polygon is 360° / n, and the interior angle is indeed 180° minus this exterior angle, which also leads to 150° for n = 12.

Final Answer:
Each interior angle of a regular dodecagon measures 150°.