The ratio between the exterior angle and the interior angle of a regular polygon is 2 : 3. Find the number of sides of the polygon.

Options

A. 10
B. 5
C. 6
D. 3

Answer

To find the number of sides of a regular polygon given the ratio of its exterior angle to its interior angle is 2:3, we can follow these steps:

### Step 1: Define the exterior and interior angles
Let the exterior angle be \( E \) and the interior angle be \( I \). According to the problem, the ratio of the exterior angle to the interior angle is given as:

\[ \frac{E}{I} = \frac{2}{3} \]

### Step 2: Express the interior angle in terms of the exterior angle
From the ratio, we can express the interior angle in terms of the exterior angle:

\[ I = \frac{3}{2}E \]

### Step 3: Use the relationship between interior and exterior angles
We know that the interior and exterior angles of a polygon are related as follows:

\[ I + E = 180^\circ \]

### Step 4: Substitute the expression for \( I \)
Substituting the expression for \( I \) into the equation:

\[ \frac{3}{2}E + E = 180^\circ \]

### Step 5: Combine like terms
Combine the terms on the left side:

\[ \frac{3}{2}E + \frac{2}{2}E = 180^\circ \]

\[ \frac{5}{2}E = 180^\circ \]

### Step 6: Solve for \( E \)
To find \( E \), multiply both sides by \( \frac{2}{5} \):

\[ E = 180^\circ \times \frac{2}{5} = 72^\circ \]

### Step 7: Find the number of sides \( N \)
The exterior angle \( E \) of a regular polygon can also be calculated using the formula:

\[ E = \frac{360^\circ}{N} \]

Setting the two expressions for \( E \) equal to each other:

\[ \frac{360^\circ}{N} = 72^\circ \]

### Step 8: Solve for \( N \)
Cross-multiply to solve for \( N \):

\[ 360^\circ = 72^\circ N \]

Dividing both sides by \( 72^\circ \):

\[ N = \frac{360^\circ}{72^\circ} = 5 \]

### Conclusion
The number of sides of the polygon is \( N = 5 \).

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