# Geometry ( find the angle )

1. Oct 5, 2012

### sambarbarian

Q. If a regular polygon has n number of sides, what will be the angles between the sides ?

i tried to do this by taking ratios of figures and angles , such as 180/3 , 360/4 and so on , but the progression is not uniform so i am clueless right now.

the answer is in terms of n and pi

2. Oct 5, 2012

### Staff: Mentor

Try looking at the exterior angles. Make a list of the regular polygons, starting with an equilateral triangle.
Code (Text):

Polygon  Exterior angle
Triangle 120°
Square ?
Pentagon ?
etc.

3. Oct 5, 2012

### HallsofIvy

Staff Emeritus
Pick one vertex and draw a line from that point to all other vertices. That will divide the n-sided polygon into n- 2 triangles. What is the total measure of the angles in those n- 1 vertices? Every angle in every triangle is part of an angle in the polygon so that total is also the total of the measures in the polygon. Since the polygon is regular, all the angles have the same measure.

For example, if the polygon is a square, it has four vertices. Choose any vertex and draw lines to the other vertices. Two of those lines are already sides of the square, the third is a diagonal. That divides the square into 4- 2= 2 triangles. Each has angle measure totaling 180 degrees so that total angle measure of the two triangles, and so of the square, is 2(180)= 360 degrees. Since there are 4 angles, and they all have the same measure, each angle has measure 360/4= 90 dergrees.

4. Oct 5, 2012

### sambarbarian

halls of ivy , i did not get you

mark , can you please explain how that is relevant to the question ?

5. Oct 5, 2012

### Staff: Mentor

Did you make the table like I suggested? If so, what did you get?

By "exterior angle" what I meant was the supplement (i.e., 180° - interior angle) of the interior angle.

6. Oct 7, 2012

### sambarbarian

square , 90
pentagon , 72
hexagon , 60 ........

7. Oct 7, 2012

### utkarshakash

It can be solved by using complex numbers.
Assuming the centre of the polygon to be at (0.0), every vertex of the polygon represents the nth root of unity which is given by

$1,α,α^{2},α^{3}..........,α^{n-1}$

Every nth root of unity represents a vertex of polygon having n sides taken anticlockwise. Now you decide how to find the argument between two sides of polygon.