Polygon Angle Relationships: Calculating Sides

This problem involves finding the number of sides of a polygon based on the relationship between its interior and exterior angles.

Key Formulas

• Sum of interior angles of an n-sided polygon: \(\\(n-2) \times 180^\circ\)
• Sum of exterior angles of any convex polygon: \(360^\circ\)

Problem Setup

Let \(n\) be the number of sides of the polygon.

The question states that the sum of the interior angles is triple the sum of the exterior angles.

We can write this relationship as an equation:

Sum of interior angles = \(3 \times\) Sum of exterior angles

Solving for Number of Sides (n)

1. Substitute the formulas into the equation: \[(n-2) \times 180^\circ = 3 \times 360^\circ\]
2. Simplify the right side of the equation: \[(n-2) \times 180^\circ = 1080^\circ\]
3. Divide both sides by \(180^\circ\) to isolate \((n-2)\): \[n-2 = \frac{1080^\circ}{180^\circ}\] \[n-2 = 6\]
4. Add 2 to both sides to solve for \(n\): \[n = 6 + 2\] \[n = 8\]

Therefore, the polygon has 8 sides.