The measure of each interior angle of a regular. polygon with \(n\) sides is \(\left[\frac{(n-2) 180}{n}\right]\) degrees. What is the measure of each interior angle of a regular polygon with \(n\) sides, in radians? A. \(\frac{(n-2) \pi}{4 n}\) B. \(\frac{(n-2) \pi}{2 n}\) C. \(\frac{(n-2) \pi}{n}\) D. \(\frac{(n-2) 2 \pi}{n}\) E. \(\frac{(n-2) 4 \pi}{n}\)

Geometric formulas are essential tools in mathematics that allow us to calculate various properties of shapes, such as their angles, sides, and areas. One of the fundamental formulas relates to the interior angles of polygons. A polygon is a closed geometric figure with straight sides. According to the formula, the sum of the interior angles of a polygon with n sides can be found using the expression \( (n-2) \times 180 \), where n is the number of sides. This formula works because a polygon can be divided into (n-2) triangles, and the sum of angles in each triangle is \(180^\circ\). Thus, for any regular polygon, the measure of a single interior angle is obtained by dividing the sum by the number of angles, leading to the formula: \( \frac{(n-2) \times 180}{n} \). Understanding this formula is key to solving numerous problems in geometry regarding polygons.

When working with angles, it's crucial to be able to switch between degrees and radians. This flexibility is important because different branches of mathematics and applications use different units. The degree to radian conversion is straightforward: there are \(2\pi\) radians in a full circle of \(360^\circ\), so to convert degrees to radians, we multiply by \(\frac{\pi}{180}\). Conversely, to convert radians to degrees, we multiply by \(\frac{180}{\pi}\). This conversion factor simplifies many calculations in trigonometry and calculus. For example, to find the radian measure of an interior angle of a regular polygon with n sides, we multiply the degree measure by the conversion factor: \(\left[\frac{(n-2) \times 180}{n}\right] \times \frac{\pi}{180}\), leading to the expression \(\frac{(n-2) \pi}{n}\), which will provide the angle in radians. Recognizing and utilizing this conversion is critical for various applications in higher mathematics.

Regular polygons are a special category of polygons where all sides and angles are equal. These properties make calculating their measurements much easier, given their symmetry. Here are a few notable properties for understanding regular polygons:

• All sides are of equal length, simplifying perimeter calculations to just the side length multiplied by the number of sides.
• The interior angles are congruent, and the formula \(\frac{(n-2) \times 180}{n}\) degrees or \(\frac{(n-2) \pi}{n}\) radians, where n is the number of sides, gives the measure of each angle.
• The exterior angles of a regular polygon always add up to \(360^\circ\), regardless of the number of sides.
• The central angle, which is the angle formed at the center of the polygon by lines drawn from the center to consecutive vertices, is \(\frac{360}{n}\) degrees or \(\frac{2\pi}{n}\) radians.

These properties not only help in solving geometry problems but also lay the groundwork for deeper exploration into shapes and their characteristics in more advanced studies.