Question

Find the number of sides of a regular polygon if it's central angle measures 35°.

Answer 1

The [# of sides (n) - 2] × 180° is the formula for the sum of interior angles (IAs) in a regular polygon. And so that sum of all IAs ÷ IA = n
Each central angle is opposite of one side.
And all of the central angles add up to 360°, therefore 35×n = 360
35n/35 = 360/35 = 10.29, but we need to verify: each 35° central angle forms an isosceles triangle with a side of the polygon, and each opposite angle (x) is 1/2 of an interior angle. All 3 angles of a triangle sum to 180°, and 2 angles (x) are equal in an isosceles, so
x + x + 35 = 180
2x + 35 = 180
2x = 145
2x/2 = 145/2
x = 72.5
And each x is 1/2 of an interior angle (IA), therefore 2x = IS
IA = 2 (72.5) = 145
(n - 2) × 180° = IA × n
(n - 2) × 180° = 145n
(n - 2) × 180°÷180 = 145n/180
n - 2 = 29n/36
36n/36 - 72-36 = 29n/36
(36n/36 - 72-36)×36 = (29n/36)×36
36n - 72 = 29n
36n - 29n = 72
7n = 72
7n/7 = 72/7
n = 10.29

Are you sure the central angle was 35, not 36°?? Because we don't get a whole number as the # of sides. How can we have 0.29 of a side?