Find the number of diagonals of a :
(i) pentagon (ii) octagon.

Answer

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

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

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

### Step-by-Step Solution:

**(i) For a Pentagon:**

1. **Identify the number of sides (n):**
A pentagon has 5 sides, so \( n = 5 \).

2. **Substitute n into the formula:**
\[ \text{Number of diagonals} = \frac{5(5-3)}{2} \]

3. **Calculate the expression inside the parentheses:**
\[ 5 - 3 = 2 \]

4. **Multiply the results:**
\[ 5 \times 2 = 10 \]

5. **Divide by 2:**
\[ \frac{10}{2} = 5 \]

Thus, the number of diagonals in a pentagon is **5**.

---

**(ii) For an Octagon:**

1. **Identify the number of sides (n):**
An octagon has 8 sides, so \( n = 8 \).

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

3. **Calculate the expression inside the parentheses:**
\[ 8 - 3 = 5 \]

4. **Multiply the results:**
\[ 8 \times 5 = 40 \]

5. **Divide by 2:**
\[ \frac{40}{2} = 20 \]

Thus, the number of diagonals in an octagon is **20**.

### Summary of Results:
- Number of diagonals in a pentagon: **5**
- Number of diagonals in an octagon: **20**

---