Sum of all interior angles of a polygon with n sides is equal to :

Question

Accepted Answer

The sum of all interior angles of a polygon with $ n $ sides is given by the formula $(n-2) \times 180^\circ$. This formula is derived from the fact that any polygon can be divided into triangles, and the sum of the interior angles of a triangle is always $180^\circ$.

### Explanation for Option 1:
1. **Derivation of the Formula**: 
   - A polygon with $ n $ sides can be divided into $(n-2)$ triangles. This is because if you take any vertex of the polygon and draw diagonals to all other non-adjacent vertices, you will form $(n-2)$ triangles.
   - Since each triangle has an interior angle sum of $180^\circ$, the total sum of the interior angles of the polygon is $(n-2) \times 180^\circ$.

2. **Example**:
   - Consider a quadrilateral ($n=4$): It can be divided into $2$ triangles. Therefore, the sum of its interior angles is $2 \times 180^\circ = 360^\circ$.
   - Consider a pentagon ($n=5$): It can be divided into $3$ triangles. Therefore, the sum of its interior angles is $3 \times 180^\circ = 540^\circ$.

### Explanation for Why Other Options are Incorrect:

- **Option 2: $\frac{n}{2} \times 180^\circ$**
  - This formula does not correctly account for the number of triangles that can be formed within a polygon. It suggests that the sum of the angles is proportional to half the number of sides, which is incorrect. For example, for a quadrilateral ($n=4$), this formula would give $360^\circ$, which coincidentally is correct, but it fails for other polygons like a pentagon ($n=5$), where it would incorrectly suggest $450^\circ$.

- **Option 3: $(n+2) \times 180^\circ$**
  - This formula incorrectly adds extra triangles to the polygon. It suggests that the sum of the angles is based on $(n+2)$ triangles, which is not possible. For a quadrilateral, this would give $1080^\circ$, which is incorrect.

- **Option 4: $\frac{n}{3} \times 180^\circ$**
  - This formula incorrectly divides the number of sides by 3, which does not relate to the number of triangles that can be formed. For a quadrilateral, this would give $240^\circ$, which is incorrect.

In conclusion, Option 1, $(n-2) \times 180^\circ$, is the correct formula for calculating the sum of all interior angles of a polygon with $ n $ sides, as it accurately reflects the number of triangles that can be formed within the polygon.