A polygon is any closed figure which is bounded by at last three straight lines. It implies that a polygon can have more than three sides, a polygon is named according to the number of sides it has thus.

Number of Sides | Name of Polygon |

3 | Triangle |

4 | Quadrilateral |

5 | Pentagon |

6 | Hexagon |

7 | Heptagon |

8 | Octagon |

9 | Nonagon |

10 | Decagon |

**Types of Polygon:**

Polygon can also be classified into the following ways

- Convex Polygon: In this type of Polygon, each interior angle is less than 180
^{o} - Reentrant Polygon: This is a polygon with one or more of the interior angles reflex i.e greater than 180
^{o} - Equivalent Polygon: All its sides are equal
- Equiangular polygon: All its angles are equal
- Regular Polygons: That is polygon in which all its sides is an angle are equal

Problems involving polygon ha to do with theorems. So we are going to walk with two theorems in order to tackle problems on it

Theorem 1:

The sum of the interior angle of any n-sided (Covex)

Polygon is given as (*2n-1)* right angle

i.e (*2n – 4) 9-*

that means

interior angles of a polygon is given by(*2n-4)90 ^{o}*

note the formula

The second says that the sum exterior angles of any convex polygon is for right angles.

i.e sum of exterior angles of a polygon = 4 right angles

= 4 x 90 = 360

Therefore, Sum of angle of any Polygon is 360

N/B: n is the number of sides of any polygon.

This this information we have obtain, we can now get into solving some problems on this topic.

**Example 1:**

The sum of the angles of a regular polygon is 2520, how many sides does the polygon

**Solution**

Sum of interior angle of regular polygon = (2n-4)90^{o}

Since the sum of the angles of a regular polygon is 2520

*(2n-4)90 ^{o}= 2520^{o}*

Let open the bracket

*2n *x* 9 – 4 *x *9 = 2520 ^{o}*

180^{o}n = 360^{o} = 2520

Collect like terms

180n = 2520^{o} + 360^{o}

180n/180 = 2880/180

N = 16

Therefore, the number of sides of the polygon is 16

**Example 2:**

One angle of an octagon is 65^{o}.The other are equal to each other find them

Solution

An octagon as 8 sides let each of the remaining 7 angles be x

Again, sum of interior angles of a regular polygon = (2n-4)90^{o}

And one of the angles is 65^{o}

This means

*7 *x*x + *65^{o} = (2n-4)90^{o}

*7x + 65 ^{o} = (2* x

*8 – 4) 90*

^{o}Let solve the right hand side first

*(16-4)90 ^{o}*

*= 12 *x *90 ^{o} = 1080^{o}*

Then, 7x + 65^{o} = 1080

Collect like terms

*7x = 1080 ^{o} = 650^{o}*

*7x/7 = 1015/7*

X = 145^{o}

Therefore, the values of the other 7 remaining angles is 145^{o}

Example 3:each angle of regular polygon is 165^{o} how many sides hat it?

Solution

each extains angle

= 180^{o} – 165^{o} = 15^{o}

Sum of angle of any polygon is 360^{o}

Solve n = 3060^{o}/exterior angle

N = 360/15 = 24

Therefore polygon has 24 sides

**Example 4:**

If the angle of a quadrilateral are x^{o} x + 10, x +20 and x + 30^{o} find the value of each angle

Solution

Quadrilateral is a polygon with 4 sides

i.e n = 4

again, sum of interior angle of a regular polygon

= (2n-4)90^{o}

That means

Sum of interior angle of a quadrilateral

=(2 x 4 – 4) x 90^{o}

-(8-4) x 90^{o}

– 4 x 90^{o} = 360

Add the angles given and equate it 3600^{o}

x + x + 12 + x + 20^{o} x + 30^{o} = 360^{o}

collect like terms

4x + 60 – 60

4x = 360 – 60

4x/4 = 300/4

X = 75^{o}

Therefore each of the angle will be 75, 75 + 10, 75 + 2, 75 + 30

75, 85, 95, and 105

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**Example 5:**

Each interior of a regular polygon is 120^{o} more than each of the exterior angles. How many sides nets it?

**Solution**

Sum of interior angles of a regular polygon =(2n-4)90^{o}

Again, Each exterior

Angle = 180 – 120 = 60

120 makes then each of the exterior angles 280 – 120 = 60

120 + Each exterior angle = 180 – 120 = 60^{o}

120 more than each of the exterior means 120 + each exterior angle means

120 + each exterior

= 120 + 60 = 180

Therefore,

(2n-4) 90^{o} = 180

180n – 360 = 180

180n = 180 + 360

180n/180 = 540/180

N = 3

Therefore, the number of sides = 3

**Example 6:**

The angle of the hexagon are y, (y+15), (y+25) (y+ 35)^{o}

(y+45) an d(y + 54) find

Solution

Sum of angles of interior

Angles of a polygon = (2n-4)90^{o}

Sum of angles of a hexagon = (2 x 6 – 4)90^{o}

(12 – 4) 90^{o}

8 x 90^{o} 720^{o}

N/B Hexagon is polygon with six sides

Add the angles and equate it to 720

Y+y+15+y+25+y+35+y

45+y+54=720

6y+174=721

6y = 720 – 174

6y/6 = 546/6

Y = TI

Therefore, the value of y is 91^{o}

In conclusion, when faced with problems on polygon, always remember the theorems and the different formulas used. Make sure you write down these formulas for reference purpose.

With the problems we have solved today, I believe polygon problems cannot be a barrier to you.