Find the circular mesure of an internal angle of a regular: (i) pentagon (ii) hexagon (iii) polygo of 40 sides.

Answer

To find the circular measure of an internal angle of a regular polygon, we can use the formula for the internal angle of a regular polygon with \( n \) sides:

\[ \text{Interior angle} = \frac{(n-2) \pi}{n} \text{ radians} \]

We will apply this formula to each part of the question.

### (i) Pentagon (5 sides)

1. **Identify the number of sides**: For a pentagon, \( n = 5 \).
2. **Substitute into the formula**:
\[ \text{Interior angle} = \frac{(5-2) \pi}{5} = \frac{3 \pi}{5} \]

**Answer**: The circular measure of an internal angle of a regular pentagon is \( \frac{3\pi}{5} \) radians.

### (ii) Hexagon (6 sides)

1. **Identify the number of sides**: For a hexagon, \( n = 6 \).
2. **Substitute into the formula**:
\[ \text{Interior angle} = \frac{(6-2) \pi}{6} = \frac{4 \pi}{6} = \frac{2 \pi}{3} \]

**Answer**: The circular measure of an internal angle of a regular hexagon is \( \frac{2\pi}{3} \) radians.

### (iii) Polygon with 40 sides

1. **Identify the number of sides**: For a polygon with 40 sides, \( n = 40 \).
2. **Substitute into the formula**:
\[ \text{Interior angle} = \frac{(40-2) \pi}{40} = \frac{38 \pi}{40} = \frac{19 \pi}{20} \]

**Answer**: The circular measure of an internal angle of a regular polygon with 40 sides is \( \frac{19\pi}{20} \) radians.

### Summary of Answers:
- (i) Pentagon: \( \frac{3\pi}{5} \) radians
- (ii) Hexagon: \( \frac{2\pi}{3} \) radians
- (iii) Polygon with 40 sides: \( \frac{19\pi}{20} \) radians