What is the number of diagonals in a
(i) heptagon (ii) octagon (iii) polygon of 12 sides ?

Answer

To find the number of diagonals in polygons, we can use the formula:

\[ \text{Number of diagonals} = \frac{n(n-3)}{2} \]

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

Let's solve the question step by step for each polygon:

### (i) Heptagon
1. **Identify the number of sides**: A heptagon has 7 sides. So, \( n = 7 \).
2. **Substitute into the formula**:
\[ \text{Number of diagonals} = \frac{7(7-3)}{2} \]
3. **Calculate \( n - 3 \)**:
\[ 7 - 3 = 4 \]
4. **Multiply \( n \) by \( n - 3 \)**:
\[ 7 \times 4 = 28 \]
5. **Divide by 2**:
\[ \frac{28}{2} = 14 \]
6. **Conclusion**: A heptagon has **14 diagonals**.

### (ii) Octagon
1. **Identify the number of sides**: An octagon has 8 sides. So, \( n = 8 \).
2. **Substitute into the formula**:
\[ \text{Number of diagonals} = \frac{8(8-3)}{2} \]
3. **Calculate \( n - 3 \)**:
\[ 8 - 3 = 5 \]
4. **Multiply \( n \) by \( n - 3 \)**:
\[ 8 \times 5 = 40 \]
5. **Divide by 2**:
\[ \frac{40}{2} = 20 \]
6. **Conclusion**: An octagon has **20 diagonals**.

### (iii) Polygon with 12 sides
1. **Identify the number of sides**: A polygon with 12 sides has \( n = 12 \).
2. **Substitute into the formula**:
\[ \text{Number of diagonals} = \frac{12(12-3)}{2} \]
3. **Calculate \( n - 3 \)**:
\[ 12 - 3 = 9 \]
4. **Multiply \( n \) by \( n - 3 \)**:
\[ 12 \times 9 = 108 \]
5. **Divide by 2**:
\[ \frac{108}{2} = 54 \]
6. **Conclusion**: A polygon with 12 sides has **54 diagonals**.

### Summary of Results
- Heptagon: 14 diagonals
- Octagon: 20 diagonals
- Polygon with 12 sides: 54 diagonals