Basic Concept of Regular Polygon

A polygon is a closed, flat figure made up of straight-line segments. A polygon is called a regular polygon if it is both:

1. Equiangular: All its interior angles are equal.

2. Equilateral: All its sides are equal in length.

Sum of interior angles = (n − 2) × 180^∘

Each interior angle = `"(n − 2) × 180° " /"n"`

Each exterior angle = `"360°"/"n"`

Number of sides (n) of a regular polygon = `"360°"/"exterior angle "`

Relationship between interior and exterior angles = Exterior angle + Interior angle = 180° 

Find each interior angle of the regular polygon with the number of sides.
(i) 20 (ii) 15

Solution:

(i) Given number of sides n = 20

Each interior angle of the 20-sided regular polygon 

                                     = `"(n − 2) × 180° " /"n"`

                                     = `"(20 − 2) × 180° " /"20"`

                                     =  `"18 × 180° " /"20"`
                                     = 162° 

(ii) Given number of sides, n = 15

Each interior angle of 15-sided regular polygon 

                                     = `"(n − 2) × 180° " /"n"`

                                     = `"(15 − 2) × 180° " /"15"`

                                     =  `"13 × 180° " /"15"`
                                     = 156°

Find each exterior angle of a regular polygon with sides:
(i) 12 (ii) 18

Solution:

If the number of sides of a regular polygon = n,

its each exterior angle = `"360°" /"n"`

(i) Number of sides, n = 12

∴ Each exterior angle = `"360°" /"n"`

                                    =  `"360°" /"12"`

                                    = 30°

(ii) Number of sides, n = 18

∴ Each exterior angle = `"360°" /"n"`

                                    =  `"360°" /"18"`

                                    = 20°

Find the number of sides of a regular polygon with each exterior angle:
(i) 18° (ii) 72°

Solution:

When exterior angle of a regular polygon is given,

the number of its sides = `"360°"/"exterior angle "`

(i) Exterior angle of the regular polygon = 18°

⇒ Number of sides in it = `"360°"/"exterior angle "`

                                       = `"360°"/"18"`

                                        =  20

(ii) Exterior angle of the regular polygon = 72° 

⇒ Number of sides in it = `"360°"/"exterior angle "`

                                       = `"360°"/"72"`

                                       =  5

Is it possible to have a regular polygon with each interior angle 135°?

Solution:

Method 1:

If n be the number of sides of the regular polygon,

∴ `"(n − 2) × 180° " /"n"`  = 135° => 180n − 360 = 135n

                                                      => 180n − 135n = 360

                                                      => 45n = 360

                                                      and,   n = `360/5` = 8

Since n = 8 is a whole number, a regular polygon with each interior angle 135° is possible.

Method 2:

Given each interior angle = 135°

and we know, at each vertex of a polygon,

interior angle + exterior angle = 180°

=>         135° + exterior angle = 180°

=>                    Exterior angle = 180° − 135°

                                                = 45°

Now, number of sides = `"360°"/"exterior angle"`

                                    = `"360°"/"45° "` = 8

Hence, a regular polygon with each interior angle of 135° is possible (an octagon).

The Natural Engineer: Bees and Hexagons

Honeybees construct their honeycombs using regular hexagonal cells. This is one of nature's most efficient designs.

Why hexagons?

• Each interior angle of a regular hexagon = 120°

• Hexagons tessellate (fit together) perfectly with no gaps

• This structure uses the least amount of wax while providing maximum storage space

• The 120° angles create strong structural integrity

Mathematical calculation:

• For a hexagon (n = 6):

• Sum of interior angles = (6 − 2) × 180° = 720°

• Each interior angle = 720° ÷ 6 = 120°

• Each exterior angle = 360° ÷ 6 = 60°

Bees instinctively create this mathematically optimal shape without any formal training in geometry!