The sum of the interior angles of a regular polygon is three times the sum of its exterior angles. Number of sides of the polygon is equal to :

Question

Accepted Answer

To determine the number of sides of a regular polygon where the sum of the interior angles is three times the sum of its exterior angles, we can use the following mathematical reasoning:

1. **Sum of Interior Angles**: The formula for the sum of the interior angles of a polygon with $ n $ sides is given by:
   $$
   \text{Sum of Interior Angles} = (n - 2) \times 180^\circ
   $$

2. **Sum of Exterior Angles**: The sum of the exterior angles of any polygon is always $ 360^\circ $.

3. **Given Condition**: According to the problem, the sum of the interior angles is three times the sum of the exterior angles:
   $$
   (n - 2) \times 180^\circ = 3 \times 360^\circ
   $$

4. **Solving the Equation**:
   $$
   (n - 2) \times 180 = 1080
   $$
   $$
   n - 2 = \frac{1080}{180}
   $$
   $$
   n - 2 = 6
   $$
   $$
   n = 8
   $$

Thus, the number of sides of the polygon is 8, which corresponds to Option 4.

**Explanation for Incorrect Options**:

- **Option 1 (5 sides)**: If the polygon had 5 sides, the sum of the interior angles would be:
  $$
  (5 - 2) \times 180 = 540^\circ
  $$
  The sum of the exterior angles remains $ 360^\circ $. Clearly, $ 540^\circ $ is not three times $ 360^\circ $.

- **Option 2 (6 sides)**: For a polygon with 6 sides, the sum of the interior angles would be:
  $$
  (6 - 2) \times 180 = 720^\circ
  $$
  Again, $ 720^\circ $ is not three times $ 360^\circ $.

- **Option 3 (7 sides)**: For a polygon with 7 sides, the sum of the interior angles would be:
  $$
  (7 - 2) \times 180 = 900^\circ
  $$
  Here, $ 900^\circ $ is not three times $ 360^\circ $.

Therefore, the only option that satisfies the condition given in the problem is Option 4, which is a polygon with 8 sides.