The ratio of sides of two regular polygon is 1:2 and ratio of their internal angle is 2:3 what is the number of sides of polygon having more sides

Options

A. 4
B. 8
C. 6
D. 12

Answer

To solve the problem, we need to find the number of sides of the polygon with more sides based on the given ratios of sides and internal angles.

### Step-by-Step Solution:

1. **Understand the Ratios Given**:
- The ratio of the sides of the two polygons is 1:2. Let the number of sides of the first polygon be \( N_1 = x \) and the second polygon be \( N_2 = 2x \).
- The ratio of their internal angles is 2:3.

2. **Formula for Internal Angles**:
- The formula for the internal angle \( A \) of a regular polygon with \( N \) sides is given by:
\[ A = \frac{180(N - 2)}{N} \]

3. **Set Up the Equation for Internal Angles**:
- For the first polygon:
\[ A_1 = \frac{180(x - 2)}{x} \]
- For the second polygon:
\[ A_2 = \frac{180(2x - 2)}{2x} \]

4. **Set Up the Ratio of Internal Angles**:
- According to the problem, the ratio of the internal angles is 2:3:
\[ \frac{A_1}{A_2} = \frac{2}{3} \]
- Substitute the expressions for \( A_1 \) and \( A_2 \):
\[ \frac{\frac{180(x - 2)}{x}}{\frac{180(2x - 2)}{2x}} = \frac{2}{3} \]

5. **Simplify the Equation**:
- Cancel out \( 180 \) and simplify:
\[ \frac{(x - 2) \cdot 2x}{x(2x - 2)} = \frac{2}{3} \]
- This simplifies to:
\[ \frac{2x(x - 2)}{x(2x - 2)} = \frac{2}{3} \]
- Further simplifying gives:
\[ \frac{2(x - 2)}{2x - 2} = \frac{2}{3} \]

6. **Cross Multiply**:
- Cross multiplying yields:
\[ 3 \cdot 2(x - 2) = 2(2x - 2) \]
- Expanding both sides results in:
\[ 6x - 12 = 4x - 4 \]

7. **Solve for \( x \)**:
- Rearranging gives:
\[ 6x - 4x = 12 - 4 \]
\[ 2x = 8 \]
\[ x = 4 \]

8. **Find the Number of Sides**:
- Now, we can find \( N_1 \) and \( N_2 \):
\[ N_1 = x = 4 \]
\[ N_2 = 2x = 2 \cdot 4 = 8 \]

9. **Conclusion**:
- The polygon with more sides has \( N_2 = 8 \) sides.

### Final Answer:
The number of sides of the polygon having more sides is **8**.