Question

Each interior angle of a regular polygon is ‘a’ times as large as each exterior angles. How many sides does the polygon have?

• A. (a + 1)
• B. (a + 2)
• C. 2(a + 1)
• D. 2(a + 2)
• E. 2(a + 4)

Answer 1

The Correct Option is C

To solve for the number of sides (\(n\)) of the regular polygon where each interior angle is 'a' times the exterior angle, we will follow these logical steps: 

1. Recall the formula for the exterior angle of a regular polygon with \(n\) sides: \(\theta_{\text{ext}} = \frac{360}{n}\).
2. Since each interior angle is 'a' times the exterior angle, the interior angle can be expressed as: \(\theta_{\text{int}} = a \cdot \theta_{\text{ext}} = a \cdot \frac{360}{n}\).
3. For any polygon, the interior and exterior angles are supplementary: \(\theta_{\text{int}} + \theta_{\text{ext}} = 180^\circ\). Substitute the expressions for both angles: \(a \cdot \frac{360}{n} + \frac{360}{n} = 180\).
4. Simplify the equation: \((a + 1) \cdot \frac{360}{n} = 180\).
5. Rearrange to solve for \(n\): \(n = \frac{360 \cdot (a + 1)}{180}\). Simplifying gives: \(n = 2(a + 1)\).

Thus, the number of sides of the polygon is \(2(a + 1)\).

Evaluate options:

• The correct answer is 2(a + 1), matching our derived expression.
• Other options such as \((a + 1)\), \((a + 2)\), \(2(a + 2)\), and \(2(a + 4)\) do not satisfy the derived calculation.