Question

what is the angle measure of each interior angle of a regular polygon with 42 sides?
a. 8.57°
b. 9.00°
c. 167.14°
d. 171.43°

Explanation:

Step1: Find the measure of each exterior angle

The sum of exterior angles of any polygon is \(360^{\circ}\).
For a regular \(n -\)sided polygon (\(n = 42\)), the measure of each exterior angle \(E=\frac{360^{\circ}}{n}\)
Substitute \(n = 42\) into the formula: \(E=\frac{360^{\circ}}{42}\approx8.57^{\circ}\)

Step2: Find the measure of each interior angle

The relationship between an interior angle \(I\) and an exterior angle \(E\) of a polygon is \(I = 180^{\circ}-E\)
Since \(E\approx8.57^{\circ}\), then \(I=180^{\circ}- 8.57^{\circ}=171.43^{\circ}\) (This is wrong, there is a calculation mistake above. Let's use the formula for interior angle directly)

The formula for the measure of an interior angle of a regular \(n -\)sided polygon is \(I=\frac{(n - 2)\times180^{\circ}}{n}\)
Substitute \(n = 42\) into the formula:

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Answer:

C. \(167.14^{\circ}\)