Maths@CSHS

A regular polygon has all sides of equal length and all angles of equal magnitude. A polygon with n sides can be divided into n triangles. 

        The angle sum of the interior angles of an n-sided polygon is given by the 

formula:     S = [180(n − 2)]^◦ = (180n − 360)^◦ 

The magnitude of each of the interior angles of an n-sided polygon is given by: x = (180n − 360)^◦⁄n

The sum of the exterior angles of a regular polygon is 360^◦ . 

Example1: The diagram below shows a regular octagon. 

a Show that x = 45. 

b Find the size of angle y.

Solution

a x = 360°÷ 8 = 45°

b BOC is isosceles

              ∧              ∧ 

   then OBC and OCB are equal   

   x° + 2y°= 180°

   45° + 2y° = 180°

   2y°= 180° – 45°

   2y°= 135°

   y° = 67.5°

Example 2:Find the sum of the interior angles of an 8-sided convex polygon (octagon). 

Solution

Use the formula x°= (180n-360)°

x°= 180×8-360 = 1080 °

                                                         QUESTIONS

1 ABCD is a square. BD and AC are diagonals which meet at O.

a Find the size of each of the angles at O.

b What type of triangle is:

2 ABCDEF is a regular hexagon.

Find the value of:

a x                b y

3 The diagram shows a tessellation of regular hexagons and equilateral triangles.

State the values of a and b and use these to explain the existence of the tessellation.

                                                        ANSWERS

 1 a 90◦ 

b i right-angled isosceles 

ii right-angled isosceles 

2 a 60◦ b 60◦  

3 a = 120, b = 60 The interior angles add up to 360◦ around a point so they tessellate.