What is the sum of all interior angles of a regular
(i) pentagon (ii) hexagon (iii) nonagon (iv) polygon of 12 sides ?

Answer

To find the sum of all interior angles of various polygons, we can use the formula:

\[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \]

where \( n \) is the number of sides of the polygon.

Let's solve the problem step by step for each polygon.

### (i) Pentagon
1. **Identify the number of sides**: A pentagon has \( n = 5 \) sides.
2. **Apply the formula**:
\[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ \]
3. **Calculate**:
\[ 3 \times 180^\circ = 540^\circ \]

### (ii) Hexagon
1. **Identify the number of sides**: A hexagon has \( n = 6 \) sides.
2. **Apply the formula**:
\[ \text{Sum of interior angles} = (6 - 2) \times 180^\circ = 4 \times 180^\circ \]
3. **Calculate**:
\[ 4 \times 180^\circ = 720^\circ \]

### (iii) Nonagon
1. **Identify the number of sides**: A nonagon has \( n = 9 \) sides.
2. **Apply the formula**:
\[ \text{Sum of interior angles} = (9 - 2) \times 180^\circ = 7 \times 180^\circ \]
3. **Calculate**:
\[ 7 \times 180^\circ = 1260^\circ \]

### (iv) Polygon of 12 sides
1. **Identify the number of sides**: A polygon with 12 sides has \( n = 12 \).
2. **Apply the formula**:
\[ \text{Sum of interior angles} = (12 - 2) \times 180^\circ = 10 \times 180^\circ \]
3. **Calculate**:
\[ 10 \times 180^\circ = 1800^\circ \]

### Summary of Results
- Sum of interior angles of a pentagon: \( 540^\circ \)
- Sum of interior angles of a hexagon: \( 720^\circ \)
- Sum of interior angles of a nonagon: \( 1260^\circ \)
- Sum of interior angles of a polygon with 12 sides: \( 1800^\circ \)