MATHS
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Polygon term originates from the two Greek words “Poly” and “Gon”, where “Poly” stands for “many” and “Gon” stands for angle. So the complete word “Polygon” deals with shapes having many angles.

Interestingly, the number of angles in a shape is always equal to the number of edges or sides it may have. In short, we can say a polygon is a shape with many sides or angles.

So, what are the various types of polygons in geometry and how are they made? Not to go so far, we use curves to make various shapes in geometry. So, to draw all polygons we use curves. Polygon has its angles, edges, vertices and diagonals. We can visualise them and see how many are they in any polygon.

Curve is a plane figure which is made by joining a number of points without lifting a pencil from paper and
without retracing any portion of drawing.

Moreover, a curve which does not intersect itself is called open curve and on the other hand, a curve which
intersect itself is called
closed curve.

Curve in geometry

Polygon is a closed curve which is made up of only finite number of
line segments
and any two line segments do not intersect each other except at their end points. It is a two dimensional figure
in geometry.

The line segments of a polygon are called sides or edges and end points of the line segments are called its
vertices or corners.

Different types of polygon

Any two sides of a polygon with a common end point are called adjacent sides of the polygon.

The end points of a side of a polygon are called adjacent vertices.

A diagonal is a line segment which is formed by joining two vertices which are not adjacent.

Example

AC is a diagonal.

BD is a diagonal.

Diagonals

A polygon is said to be a convex polygon, if its diagonals completely lie inside of it. Its all interior angles are less than 180 degree.

Convex Polygon

A polygon is said to be a concave polygon, if its diagonals completely lie outside of it. Its one or more than one interior angle is more than 180 degrees.

Concave Polygon

A regular polygon is a polygon whose all sides and all angles are equal. In other words we can say, it is both equilateral and equiangular.

Example

Square is a regular polygon as its sides are equal and all angles are equal.

Regular Polygon

Formula

Interior angle of regular polygon

= (n-2) × 180°

or

= (n-2) × π radian

Polygon which is not a regular polygon, i.e. if it is not equiangular and not equilateral is called irregular polygon.

Example

Rectangle is an irregular polygon.

Irregular Polygon

It is a flat shape consisting of straight and non intersecting line segments that are joined to form a single
close path. A Simple Polygon does not cross over itself.
It is also called Jordan Polygon.

All convex polygons are simple.

Simple Polygon

Number of sides of polygon | Name of polygon | Number of angles of polygon |
---|---|---|

3 | Triangle | 3 |

4 | Quadrilateral | 4 |

5 | Pentagon | 5 |

6 | Hexagon | 6 |

7 | Heptagon | 7 |

8 | Octagon | 8 |

9 | Nonagon | 9 |

10 | Decagon | 10 |

A closed plane figure formed by line segments and two line segments which intersect at only their end points.

The sum of angles of polygon with n sides is (n-2)180^{0}.

Sum of exterior angles of polygon is 360^{0}.

Yes, equilateral triangle is a regular polygon as its all sides are equal and all its angles are equal.

No, rectangle is not a regular polygon as its angles are equal but all sides are not equal.

**Solution**

Let number of sides of polygon = n

Each interior angle = $\frac{\mathrm{(n-2)180}}{\mathrm{n}}$

140 = $\frac{\mathrm{(n-2)180}}{\mathrm{n}}$

140n = 180n - 360

360 = 180n - 140n

360 = 40n

$\frac{360}{40}$ = n

n = 9

**Solution**

Sum of all exterior angle = 360^{0}

Each exterior angle = 40^{0}

Number of exterior angles = $\frac{360}{40}$

Number of exterior angles = 9

∴ Number of sides of polygon = 9

**Solution**

Each interior angle = 150^{0}.

We know that, interior angles + exterior angles = 180^{0}

150^{0} + exterior angles = 180^{0}

exterior angles = 180^{0} - 150^{0}

exterior angles = 30^{0}

**Solution**

Let exterior angle and interior angle be x and 5x respectively.

Also, interior angles + exterior angles = 180^{0}

5x + x = 180^{0}

6x = 180^{0}

x = $\frac{180}{6}$

x = 30^{0}

∴ exterior angle = 30^{0}

interior angle = 5x = 5 × 30^{0}

i.e. interior angle = 150^{0}