To find the number of sides of a regular polygon whose interior angle is twice the exterior angle, we can follow these steps: ### Step 1: Understand the Relationship Between Interior and Exterior Angles 1. **Exterior Angle**: The exterior angle of a polygon is the angle formed between one side of the polygon and the extension of an adjacent side. For a regular polygon, all exterior angles are equal. 2. **Interior Angle**: The interior angle is the angle formed inside the polygon at each vertex. For a regular polygon, all interior angles are equal. The relationship between the interior angle (I) and the exterior angle (E) of a polygon can be expressed as: \[ I + E = 180^\circ \] ### Step 2: Set Up the Equation Based on the Problem Statement According to the problem, the interior angle is twice the exterior angle: \[ I = 2E \] ### Step 3: Substitute the Interior Angle in Terms of the Exterior Angle Using the relationship \( I + E = 180^\circ \), we can substitute \( I \): \[ 2E + E = 180^\circ \] \[ 3E = 180^\circ \] ### Step 4: Solve for the Exterior Angle Now, we can solve for \( E \): \[ E = \frac{180^\circ}{3} = 60^\circ \] ### Step 5: Find the Number of Sides of the Polygon The formula for the exterior angle of a regular polygon with \( n \) sides is: \[ E = \frac{360^\circ}{n} \] Setting this equal to the exterior angle we found: \[ \frac{360^\circ}{n} = 60^\circ \] Now, we can solve for \( n \): \[ n = \frac{360^\circ}{60^\circ} = 6 \] ### Conclusion The number of sides of the regular polygon is **6**. ### Explanation of Other Options - **Option B (2 sides)**: A polygon with 2 sides is not a valid polygon in Euclidean geometry. The minimum number of sides for a polygon is 3 (a triangle). - **Option C (3 sides)**: A triangle has an exterior angle of \( 120^\circ \) (since \( \frac{360^\circ}{3} = 120^\circ \)), which does not satisfy the condition that the interior angle is twice the exterior angle (the interior angle would be \( 60^\circ \)). - **Option D (8 sides)**: An octagon has an exterior angle of \( 45^\circ \) (since \( \frac{360^\circ}{8} = 45^\circ \)). The interior angle would be \( 135^\circ \), which is not twice the exterior angle. ### Revision Summary - The interior angle of a polygon is related to the exterior angle by the equation \( I + E = 180^\circ \). - If the interior angle is twice the exterior angle, we can set up the equation \( 2E + E = 180^\circ \) to find \( E \). - The number of sides \( n \) can be calculated using the formula \( E = \frac{360^\circ}{n} \). - For this problem, the correct answer is that the polygon has **6 sides**.