Show that centeral angle of n-sided regular polygon is 360/n

divide the n-gon into n triangles. Because the n-gon is regular, all the triangles are congruent. Thus, there are n equal angles which add up to 360°. So, each central angle is 360/n.

To show that the central angle of an n-sided regular polygon is 360/n, we can use the following steps:

Step 1: Draw the regular polygon.
Start by drawing a regular polygon with n sides. Each side should have the same length, and all the angles should be the same.

Step 2: Identify the central angle.
Next, identify the central angle of the polygon. The central angle is the angle formed by two consecutive radii (lines drawn from the center of the polygon to two adjacent vertices).

Step 3: Determine the sum of the central angles.
Since the central angle of a regular polygon is the same for all vertices, we need to find the sum of all the central angles. To do this, we can start from one vertex and go around the polygon by consecutive vertices until we reach the starting point again. This will cover all n vertices and the corresponding central angles.

Step 4: Calculate the sum of the central angles.
We know that the sum of all the angles in a polygon with n sides is (n-2) * 180 degrees. Since each central angle is the same, we can divide this sum by n to get the measure of one central angle:
Sum of central angles = (n-2) * 180 degrees
One central angle = Sum of central angles / n = (n-2) * 180 degrees / n

Step 5: Simplify the expression.
To simplify the expression, we can distribute the division:
One central angle = (n * 180 degrees - 2 * 180 degrees) / n
One central angle = (180 degrees * (n - 2)) / n
One central angle = 360 degrees * (n - 2) / n

Step 6: Finalize the answer.
We now have the measure of one central angle in terms of n. Simplifying the expression further:
One central angle = 360 degrees * (n - 2) / n
One central angle = 360 degrees * n/n - 360 degrees * 2/n
One central angle = 360 degrees - 720 degrees / n
One central angle = 360 degrees/n

Thus, we have shown that the central angle of an n-sided regular polygon is 360/n.