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

The correct answer is option (C):

2(a + 1)

Let's break down this geometry problem step-by-step.

First, recall some key facts about regular polygons:

1. The sum of the interior angles of an n-sided polygon is (n-2) * 180 degrees.
2. Each interior angle of a regular n-sided polygon is [(n-2) * 180] / n degrees.
3. The sum of the exterior angles of any polygon is always 360 degrees.
4. Each exterior angle of a regular n-sided polygon is 360 / n degrees.

Now, let's represent the given information in equations. We are told that the interior angle is 'a' times as large as the exterior angle. Using the formulas above:

Interior angle = a * Exterior angle

[(n-2) * 180] / n = a * (360 / n)

To solve for 'n' (the number of sides), let's simplify and manipulate the equation:

Multiply both sides by n:

(n-2) * 180 = a * 360

Divide both sides by 180:

n - 2 = 2a

Add 2 to both sides:

n = 2a + 2

Factor out a 2 from the right side:

n = 2(a + 1)

Therefore, the polygon has 2(a + 1) sides. This matches one of the provided options.