Find the number of sides in a regular polygon, if its each exterior angle is :
`(i) 1/3` of a right angle
(ii) two-fifths of a right angle

Answer

To find the number of sides in a regular polygon based on the given exterior angles, we can use the formula for the exterior angle of a regular polygon:

\[ \text{Exterior angle} = \frac{360^\circ}{n} \]

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

### Part (i): Each exterior angle is \( \frac{1}{3} \) of a right angle

1. **Convert the angle**: A right angle is \( 90^\circ \). Therefore, \( \frac{1}{3} \) of a right angle is:
\[ \frac{1}{3} \times 90^\circ = 30^\circ \]

2. **Set up the equation**: We can equate the exterior angle to \( 30^\circ \):
\[ \frac{360^\circ}{n} = 30^\circ \]

3. **Cross-multiply to solve for \( n \)**:
\[ 360^\circ = 30^\circ \times n \]
\[ n = \frac{360^\circ}{30^\circ} \]

4. **Calculate \( n \)**:
\[ n = 12 \]

Thus, the number of sides in the polygon for part (i) is **12**.

### Part (ii): Each exterior angle is \( \frac{2}{5} \) of a right angle

1. **Convert the angle**: A right angle is \( 90^\circ \). Therefore, \( \frac{2}{5} \) of a right angle is:
\[ \frac{2}{5} \times 90^\circ = 36^\circ \]

2. **Set up the equation**: We can equate the exterior angle to \( 36^\circ \):
\[ \frac{360^\circ}{n} = 36^\circ \]

3. **Cross-multiply to solve for \( n \)**:
\[ 360^\circ = 36^\circ \times n \]
\[ n = \frac{360^\circ}{36^\circ} \]

4. **Calculate \( n \)**:
\[ n = 10 \]

Thus, the number of sides in the polygon for part (ii) is **10**.

### Summary of Solutions:
- For part (i), the number of sides is **12**.
- For part (ii), the number of sides is **10**.

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