# Polygons

A polygon is a plane figure bounded by three or more line segments. It is said to be:

a) equilateral when all its sides all equal;

b) equiangular, when all its angles are equal;

c) regular, when all its sides and all its angles are equal.

# Different types of polygons

Sides |
Name |

3 |
Triangle |

4 |
Quadrilateral |

5 |
Pentagon |

6 |
Hexagon |

7 |
Heptagon |

8 |
Octagon |

9 |
Nonagon |

10 |
Decagon |

Infinity |
Circle |

# Similar Polygons

Two polygons are said to be similar to each other ifi) Their corresponding angles are equal

ii) The lengths of their corresponding sides are proportional.

**Points to remember: **

- In a convex polygon (a polygon in which none of its interior angles is more than 180Â°) of 'n' sides, the sum of the interior angles is equal to (2n - 4) right angles.
- Each interior angle of a regular polygon = (360/number of sides)

If we produce a side of polygon, the angle it makes with the next side is called an exterior angle. In the diagram given below, ABCDE is a pentagon. Its side AB has been produced to P, then âˆ CBP is an exterior angle.

Also, as an exterior angle and its adjacent interior angle make a straight line, we have an exterior angle + adjacent interior angle = 180Â°

- Each interior angle = 180Â°- exterior angle
- Number of diagonals of a polygon of n sides = n(n-1) - n
- Area = 1/2 (perimeter) (perpendicular from centre to any side).

**Diagonal of a polygon:**

Line segment joining any two non-consecutive vertices of a polygon is called its diagonal.

Convex polygon: If all the (interior) angles of a polygon are than 180Â°, it is called a convex polygon. In the figure given below, ABCDEF is a convex polygon.

**Concave Polygon:**

If one or more of the (interior) angles of a polygon is greater than 180Â° i.e. reflex, it is called

**Regular polygon:** A polygon is called regular polygon if all its sides have equal length and all its angles have equal size.

Thus, in a regular polygon

(i) all sides are equal in length

(ii) all interior angles are equal in size

(iii) all experior angles are equal size

**Note: **All regular polygons are convex. * *

# Special types of polygons

**Quadrilaterals:**

In a quadrilateral, the sum of all four angles is 360o.

Area = 1/2 (one diagonal) (sum of perpendiculars on it from opposite vertices) Straight lines joining the midpoints of the adjacent sides of any quadrilateral forms a parallelogram.

Squar:

A square is quadrilateral which has all sides equal, the diagonals are also equal and bisect each other at right angles. Area: side2 Perimeter: 4 (side) Diagonal: âˆš2 (side)2

Rectangle:

A rectangle is a quadrilateral with opposite sides equal and its angles are 90o each. The diagonals are equal and bisect each other but NOT at right angles. Area: length Ã— breadth Perimeter: 2 length + 2 breadth. Diagonal: âˆš length2 + breadth2

**
Rhombus:** A rhombus has all sides equal, its opposite sides are parallel the diagonals bisect each other at right angles but are not equal. Area: 1/2 (d1 Ã— d2)

** **

**Parallelogram:** A parallelogram has opposite sides parallel and equal, its opposite angles are equal and any two adjacent angles are supplementary. The diagonals bisect each other and each diagonal divides the parallelogram into two equal triangles. It cannot be inscribed in a circle. Area: base Ã— height

** **

**Trapezium:** A trapezium has two opposite sides parallel but the other two sides are not parallel. The median (line joining the midpoints of the non-parallel sides) is half the sum of parallel sides. An isosceles trapezium is one if which the non-parallel sides are equal. Area: 1/2 (sum of parallel sides) Ã— height