How many sides has a regular polygon whose interior angle is 135°?

Analysis

To find the number of sides of a regular polygon when the interior angle is given, we use the relationship between interior and exterior angles.

Concept

1. Interior + Exterior Angle = 180°: In any polygon, the interior angle and its corresponding exterior angle lie on a straight line, so they sum up to 180°.
2. Sum of Exterior Angles: The sum of all exterior angles of any convex polygon is always 360°.
3. Formula for Exterior Angle: For a regular polygon with $n$ sides, each exterior angle is given by: $$\text{Exterior Angle} = \frac{360^\circ}{n}$$

Step-by-Step Solution

Step 1: Find the exterior angle. Since the interior angle is 135°, we calculate the exterior angle:

Step 2: Use the exterior angle formula to find $n$ (the number of sides). Using the formula $\text{Exterior Angle} = \frac{360^\circ}{n}$:

Step 3: Solve for $n$. Multiply both sides by $n$:

Divide by 45:

The polygon has 8 sides (it is an octagon).

Conclusion

Comparing our result with the options provided, the correct option is D.

Exam Tip

Whenever you are given the interior angle, always convert it to the exterior angle first. It is much easier to divide 360 by the exterior angle than to use the more complex interior angle formula: $\text{Interior Angle} = \frac{(n-2) \times 180}{n}$. Saving time on calculations helps you finish your JAMB paper faster!