How many diagnals are there in a polygen of (i) 8 sides (ii) 10 sides?

Answer

To find the number of diagonals in a polygon, we can use the formula for the number of diagonals in a polygon with \( n \) sides, which is given by:

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

Now, let's solve the problem step by step for both parts.

### Part (i): Polygon with 8 sides

1. **Identify the number of sides (n)**:
Here, \( n = 8 \).

2. **Substitute n into the diagonal formula**:
\[ \text{Number of Diagonals} = \frac{8(8-3)}{2} \]

3. **Calculate the expression**:
\[ = \frac{8 \times 5}{2} \]
\[ = \frac{40}{2} = 20 \]

4. **Conclusion for part (i)**:
Thus, the number of diagonals in an 8-sided polygon is **20**.

### Part (ii): Polygon with 10 sides

1. **Identify the number of sides (n)**:
Here, \( n = 10 \).

2. **Substitute n into the diagonal formula**:
\[ \text{Number of Diagonals} = \frac{10(10-3)}{2} \]

3. **Calculate the expression**:
\[ = \frac{10 \times 7}{2} \]
\[ = \frac{70}{2} = 35 \]

4. **Conclusion for part (ii)**:
Thus, the number of diagonals in a 10-sided polygon is **35**.

### Summary of Results:
- For an 8-sided polygon, the number of diagonals is **20**.
- For a 10-sided polygon, the number of diagonals is **35**.