The sum of the angles in a polygon with \( n \) sides is given by \( (n-2) \times 180^\circ \). If a polygon has 9 sides, what is the measure of each interior angle if the polygon is regular?

["The Sum of Interior Angles in a 9-Sided Regular Polygon", "Understanding the sum of interior angles in polygons is essential in geometry, especially when exploring regular polygons. For any polygon with ( n ) sides, the formula ( (n - 2) \ imes 180^\circ ) gives the total sum of its interior angles. This formula stems from dividing a polygon into triangles and leveraging their internal angle properties.", "## How the Formula Works", "A polygon with ( n ) sides can be divided into ( n - 2 ) triangles. Each triangle has interior angles summing to ( 180^\circ ). Therefore, the total sum of all interior angles in the polygon is:", "[\n\ ext{Sum of interior angles} = (n - 2) \ imes 180^\circ\n]", "For a regular polygon—where all sides and angles are equal—the total sum is evenly distributed among the ( n ) angles. Thus, the measure of each interior angle in a regular polygon is simply the total divided by ( n ):", "[\n\ ext{Measure of each interior angle} = \frac{(n - 2) \ imes 180^\circ}{n}\n]", "## Applying the Formula to a 9-Sided Polygon", "Now, consider a regular polygon with ( n = 9 ) sides—commonly known as a nonagon. Applying the formula:", "[\n\ ext{Sum of interior angles} = (9 - 2) \ imes 180^\circ = 7 \ imes 180^\circ = 1260^\circ\n]", "Since all angles are equal in a regular polygon, each interior angle measures:", "[\n\ ext{Each interior angle} = \frac{1260^\circ}{9} = 140^\circ\n]", "## Why This Matters", "This concept is vital in architecture, engineering, art, and design—where precise angle measurements ensure structural integrity and aesthetic appeal. Knowing that each interior angle in a regular nonagon is ( 140^\circ ) helps in constructing intricate patterns, tiles, or polygonal frameworks.", "## Conclusion", "The sum of interior angles in a regular ( n )-sided polygon is ( (n - 2) \ imes 180^\circ ). For a regular nonagon (9 sides), this yields a total interior angle sum of ( 1260^\circ ), with each interior angle measuring ( 140^\circ ). This clear and scalable formula enables accurate calculations across countless geometric applications."]