Question

the exterior angle of a regular polygon measures $5^{circ}$. how many sides does the polygon have?

Explanation:

Step1: Recall the formula for exterior angles of a regular polygon

The sum of the exterior angles of any regular polygon is always \(360^\circ\). Let \(n\) be the number of sides of the regular polygon. If each exterior angle is \(5^\circ\), then we can use the formula: \(\text{Measure of each exterior angle}=\frac{\text{Sum of exterior angles}}{n}\)

Step2: Solve for \(n\)

We know the sum of exterior angles is \(360^\circ\) and each exterior angle is \(5^\circ\). So we can set up the equation \(5^\circ=\frac{360^\circ}{n}\). To solve for \(n\), we can rearrange the equation to \(n = \frac{360^\circ}{5^\circ}\)

Step3: Calculate the value of \(n\)

\(n=\frac{360}{5}=72\)

Answer:

The polygon has \(\boldsymbol{72}\) sides.