Question

Calculate the interior angles of a regular polygon with (a)20 sides (b)10 sides(c)11 sides(d)12sides
In the question above calculate each of the exterior angles

Answer 1

if you walk clockwise around outside a regular polygon of n sides you end up turning right 360 degees.
each right turn is (1/n)360 = 360 / n = exterior turn angle
each interior angle is 180 - the exterior angle = 180 - (360/n)
= (180 n -360) / n = 180 (n - 2)/n

Answer 2

(a) For a 20-sided polygon:
Each interior angle of a regular polygon with n sides is given by:
I = (n-2) x 180 / n
So for a 20-sided polygon:
I = (20-2) x 180 / 20
I = 162 degrees

Each exterior angle of a regular polygon is equal to 360 degrees divided by the number of sides:
E = 360 / n
So for a 20-sided polygon:
E = 360 / 20
E = 18 degrees

(b) For a 10-sided polygon:
Each interior angle of a regular polygon with 10 sides is given by:
I = (10-2) x 180 / 10
I = 144 degrees

Each exterior angle of a regular polygon with 10 sides is equal to:
E = 360 / 10
E = 36 degrees

(c) For an 11-sided polygon:
Each interior angle of a regular polygon with 11 sides is given by:
I = (11-2) x 180 / 11
I = 147.27 degrees

Each exterior angle of a regular polygon with 11 sides is equal to:
E = 360 / 11
E = 32.73 degrees

(d) For a 12-sided polygon:
Each interior angle of a regular polygon with 12 sides is given by:
I = (12-2) x 180 / 12
I = 150 degrees

Each exterior angle of a regular polygon with 12 sides is equal to:
E = 360 / 12
E = 30 degrees

Answer 3

Thank you for adding the additional explanation! You are correct, this formula can also be derived using the fact that the sum of the exterior angles of a polygon is always 360 degrees. Therefore, each exterior angle of a regular polygon with n sides is 360/n degrees. Since the interior and exterior angles of a polygon are supplementary (they add up to 180 degrees), we can express the interior angle I as:

I = 180 - E
= 180 - (360/n)
= 180n/n - 360/n
= (180n - 360)/n
= 180(n-2)/n

This is the same formula we derived earlier using the formula for the interior angle of a regular polygon. Thank you for pointing out this alternative method!