Question

determine for which n the number of degrees in an interior angle of a regular n-gon is a positive integer

Answer 1

Note: n- Gon is a polygon, which is having n number of sides. Poly means many and Gon means angle..

1.square has 4 sides, sum of interior angles 360degrees

2.pentagon has 5 sides, the sum of interior angles 540 degrees....etc

Regular polygon: is having all the same values of angles

Step 1:

The formula for the sum of the interior angles of an n-gon is given by

Each interior angle value is

Step 2: To determine for which 'n', the 'm' should be a positive integer.

m is the positive integer if and only if , is an integer

   is an integer if and only if n must be a factor of 360

Step 3:

We can write 360=2³.3².5¹

n is a factor of 360, so is of the form n= 2^a.3^b.5^c

where

total number of possibilities =4

total number of possibilities =3

total number of possibilities =2

The product of the different number of values n=4.3.2=24

Step 4:

but for below 2 combinations, the total angle of polygon 180-(360/n) is

for n=2⁰.3⁰.5⁰=1,180-(360/n) =negative (not a polygon)

for n=2¹.3⁰.5⁰=2, 180-(360/n) =0(not a polygon)

step 5:

therefore subtract the total number of sides from 2

n=24-2=22

n=22

step 6:

Result:

Therefore,

For n=22 the number of degrees in interior angle of a regular n-gon is a positive integer