The sum of the interior angles of a polygon is twice the sum of its exterior angles taken one at each vertex. How many sides does the polygon have?

Introduction / Context:
This question tests concepts from basic geometry about polygons, specifically the relationship between interior angles, exterior angles, and the number of sides. Understanding the standard formulas for the sum of interior angles and the sum of exterior angles of a polygon is crucial for solving many geometry problems that appear in aptitude and competitive exams.

Given Data / Assumptions:
- Let the polygon have n sides.- The sum of its interior angles is twice the sum of its exterior angles taken one at each vertex.- The polygon is assumed to be simple and convex so that the usual angle sum formulas apply.

Concept / Approach:
Two key geometric facts are used here. First, for an n sided polygon, the sum of interior angles is given by (n - 2) * 180 degrees. This arises by dividing a polygon into (n - 2) triangles. Second, for any convex polygon, the sum of the exterior angles, taking one exterior at each vertex, is always 360 degrees, regardless of the number of sides. The problem states that the sum of the interior angles is twice the sum of the exterior angles, so we can set up an equation using these formulas and solve for n.

Step-by-Step Solution:
- Let n be the number of sides of the polygon.- Sum of interior angles = (n - 2) * 180 degrees.- Sum of exterior angles (one at each vertex) = 360 degrees.- Given condition: sum of interior angles = 2 * sum of exterior angles.- Substitute the formulas: (n - 2) * 180 = 2 * 360.- Simplify the right side: 2 * 360 = 720.- So we have (n - 2) * 180 = 720.- Divide both sides by 180: n - 2 = 720 / 180.- Compute the division: 720 / 180 = 4.- Hence n - 2 = 4, so n = 4 + 2 = 6.

Verification / Alternative check:
Check the result for n = 6. A hexagon has interior sum (6 - 2) * 180 = 4 * 180 = 720 degrees. The sum of exterior angles is 360 degrees. Twice the sum of exterior angles is 2 * 360 = 720 degrees, which matches the interior sum. Therefore, the condition is satisfied exactly when n = 6, confirming the correctness of the answer.

Why Other Options Are Wrong:
- Option 4: For n = 4, the interior sum is (4 - 2) * 180 = 360 degrees, which equals the exterior sum, not twice.- Option 5: For n = 5, interior sum is (5 - 2) * 180 = 540 degrees, which is not equal to 720 degrees.- Option 7: For n = 7, interior sum is (7 - 2) * 180 = 900 degrees, which is more than twice 360 degrees.- Option 8: For n = 8, interior sum is (8 - 2) * 180 = 1080 degrees, again not twice the exterior sum.

Common Pitfalls:
Learners sometimes mistakenly think that the sum of the exterior angles equals the sum of the interior angles, or they confuse interior and exterior measures at each vertex. It is important to remember that the total of all exterior angles is fixed at 360 degrees for any convex polygon, while the interior sum grows linearly with the number of sides. Whenever a question relates interior and exterior sums, carefully substitute the correct formulas rather than guessing. Writing down the equation step by step helps avoid accidental algebra errors.

Final Answer:
The polygon has 6 sides.