Understanding Polygon Angle Properties

For any regular polygon, the sum of an interior angle and its corresponding exterior angle at any vertex is always $180^\circ$.

Let the interior angle be represented by $I$ and the exterior angle by $E$. We know:

• $I + E = 180^\circ$

Using the Given Angle Ratio

The question states that the ratio of the interior angle to the exterior angle is $4:1$. We can represent this relationship:

• $\frac{I}{E} = \frac{4}{1}$
• This implies $I = 4E$.

Calculating the Exterior Angle

Substitute the expression for $I$ ($I = 4E$) into the equation $I + E = 180^\circ$:

• $4E + E = 180^\circ$
• $5E = 180^\circ$
• $E = \frac{180^\circ}{5}$
• $E = 36^\circ$

So, the measure of each exterior angle is $36^\circ$.

Calculating the Number of Sides

The formula for the measure of an exterior angle of a regular polygon with $n$ sides is:

• $E = \frac{360^\circ}{n}$

Now, substitute the calculated value of $E$ ($36^\circ$) into the formula:

• $36^\circ = \frac{360^\circ}{n}$

Solve for $n$ (the number of sides):

• $n = \frac{360^\circ}{36^\circ}$
• $n = 10$

Therefore, the regular polygon has 10 sides.