Calculate the interior angle of a 5 - sided regular polygon

Analysis

The question asks for the measure of a single interior angle of a regular polygon with 5 sides (a pentagon).

Concept

To find the interior angle of any regular polygon, we use two main steps:

1. Find the sum of interior angles: The formula for the sum of interior angles of a polygon with $n$ sides is $(n - 2) \times 180^\circ$.
2. Find a single interior angle: Since it is a regular polygon (all angles are equal), we divide the sum by the number of sides ($n$): $$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$$

Solution

Given that $n = 5$:

Step 1: Calculate the sum of the interior angles.

Step 2: Calculate the measure of one interior angle.

Alternatively, you can find the exterior angle first:

Since the interior and exterior angles lie on a straight line, they sum to $180^\circ$:

Conclusion

The calculation confirms that the interior angle of a regular pentagon is $108^\circ$.

Exam Tip

For regular polygons, it is often faster to calculate the exterior angle first ($\frac{360}{n}$) and subtract it from $180^\circ$ than to calculate the total sum of interior angles. This saves time and reduces the risk of calculation errors!