Use the angle of the sun of triangles to develop the interior angle sums of regular polygons formula using decomposition test the formula by finding the sum of the interior angles of a square
To develop the formula for the interior angle sums of regular polygons using the angle of the sun of triangles, we can start by considering a regular polygon with n sides.
1. Divide the polygon into n congruent triangles by drawing lines from one vertex to all other vertices. Each triangle will have a central angle, which is equal to the interior angle of the polygon since all triangles formed are congruent.
2. The sum of the interior angles of a triangle is 180 degrees. Therefore, the sum of the n central angles (or the interior angles of the polygon) is equal to n * 180 degrees.
3. Using the fact that the exterior angle of a polygon is equal to the supplement of the interior angle, we can calculate the measure of each exterior angle of the polygon. Since a full rotation is 360 degrees, each exterior angle is (360 - interior angle).
4. The sum of all exterior angles of a polygon is always 360 degrees. If the measure of each exterior angle is (360 - interior angle), the sum of all exterior angles can be given as n * (360 - interior angle).
5. The sum of the interior angles and the sum of the exterior angles of a polygon are equal (as both sums cover the entire rotation around a point). Therefore, the expression calculated in step 4 should be equal to the expression calculated in step 2.
Equating the expressions for the sum of the interior angles and the sum of the exterior angles, we get:
n * (360 - interior angle) = n * 180
Simplifying this equation, we find:
360 - interior angle = 180
-interior angle = 180 - 360
-interior angle = -180
interior angle = 180 degrees
So, the sum of the interior angles of a regular polygon is always equal to 180 degrees.
Now, let's test this formula by finding the sum of the interior angles of a square:
A square has four sides, so using the formula:
Sum of interior angles = number of sides * 180 degrees
Sum of interior angles = 4 * 180 degrees
Sum of interior angles = 720 degrees
Hence, the sum of the interior angles of a square is 720 degrees, which confirms the validity of the formula.