Download In a Regular Polygon

POLYGON
When we study about shapes, we come across with mainly two types of shapes(1) Geometrical Shapes
(2) Non-geometrical Shapes
Geometrical shapes are the predefined shapes in geometry, while non-
geometrical shapes may be any arbitrary shape. Geometrical shapes may include
square, rectangle, triangle, quadrilateral, parallelogram, rhombus, cube cuboid,
pyramid, cone, sphere, circle and many more.
Polygon is also a very important geometrical figure. A two-dimensional closed
figure bounded by three or more line segments is called as a polygon. The line
segments forming a polygon are called its sides. It has three or more sides and
three or more interior angles. The point of intersection of two consecutive sides of
a polygon is called as the vertex of the polygon. The number of vertices of the
polygon is equal to the number of sides. A line segment joining any two nonconsecutive vertices is called its diagonal.
Polygons are classified on the basis of number of sides (1) Triangle (polygon containing 3 sides)
(2) Quadrilateral (polygon containing 4 sides)
(3) Pentagon (polygon containing 5 sides)
(4) Hexagon (polygon containing 6 sides)
(5) Heptagon or Heptagon (polygon containing 7 sides)
(6) Octagon (polygon containing 8 sides)
(7) Nenagon (polygon containing 9 sides)
(8) Decagon (polygon containing 10 sides)
(9) Hendecagon (polygon containing 11 sides)
(10) Dodecagon (polygon containing 12 sides)
and so on...
A sample image of a polygon is shown below -
Polygon Definition
Polygon is a two-dimensional shape bounded with straight lines. These straight
lines are called its edges, and the points where two edges meet are the vertices of
the polygon.
Triangles, Rectangles, Pentagons and Octagon all are polygons.
Properties of Polygons
In a convex polygon:
1. Polygon sum conjecture: Sum of all the interior
angles of a polygon of n sides = (n - 2) * 1800 .
2. Sum of all the exterior polygon angles of a
polygon of n sides = 4 * 900 .
3. At each vertex of a polygon, Interior Angle +
Exterior Angle = 1800 .
In a Regular Polygon:
1. Polygon interior angles of a regular polygon of
n sides = (2n−4)∗90on.
2. Each exterior angle of a regular polygon of n
sides = 360on.
3. If each exterior angle of a regular polygon is
x0, then number of side of polygon = 360ox.
4. Number of diagonals in polygon of n sides is
[n(n - 1)/2 - n].
Polygon Formulas
The area of a polygon tells how many square units are needed to cover the
polygon. Area can be measured in different units like square meter, square feet
etc. Areas of some polygons are given below:
Polygon
Triangle
Area of Polygon
a+b+c
12 Base * Height
Rectangle
Length * Width
Square
(Side)2
Trapezoid
Perimeter of Polygon
12 (Sum of parallel
sides) * Height
Parallelogram Base * Height
(where, a, b and c are the sides of the
triangle)
2(a + b)
(where, a = Length and b = width)
4 Side
a+b+c+d
where, a, b, c and d are the lengths of each
side of a trapezoid
2(a + b)
Where, a is the base and b is the other side
of the parallelogram.
Types of Polygons
There are number of polygons based on the number of sides. Polygons are named
according to the number of their sides.
Number
3
4
5
6
7
8
9
10
of sides
Polygon
Triangle quadrilateral Pentagon Hexagon Septagon Octagon Nonagon Decagon
Names
Polygons are also classified into three different types based on their angles. They
are,

Convex Polygon

Concave Polygon

Regular Polygon
Convex Polygon
If each angle of a polygon is less than 180, then it is called as a convex polygon.
In the above figure, ABCDE is a Convex Polygon.
Concave Polygon
If at least one angle of a polygon is a reflex angle i.e. more than 1800, then it is
called as a concave polygon.
In the above figure, ABCDE is a Concave Polygon. Angle DOE is the reflex angle
i.e. more than 1800.
Regular Polygon
In the above figure, ABCDE is a regular polygon. All the angles and sides of this
polygon are equal.
Identifying Polygons
Polygons can be described as 2-dimension closed figures with 3 or more sides.
Polygons are classified and described by the number of sides and by the kind of
angles.
For example, we identified

3-sided polygon as a triangle

4-sided polygon as a quadrilateral

5-sides polygon as a pentagon.
Triangle:
The triangle is a basic shape of geometry. The sum of the angles of a triangle will
be equal to 180 degree. Triangles are of 6 types. They are,






Right triangles
Acute triangle
Obtuse triangle
Equilateral triangle
Isosceles triangle
Scalene triangle
.
Quadrilateral:
The polygon has 4 sides and 4 vertices is called Quadrilateral polygon. Sum of the
angle of a quadrilateral is 360 degree. Quadrilaterals are of 4 types. They are

Square

Rectangle

Rhombus


Parallelogram
Trapezoid
Pictures of Polygons
Below picture shows the types of polygons:
Pentagon Polygon
Pentagon polygon has 5 sides and 5 vertices. The sum of the angle of a pentagon
is 540 degree.
There are two types of polygons:


Regular Pentagon
Irregular pentagon
Let us see the pentagon figure:
Regular Polygons
A polygon having all sides equal and all interior and exterior angles equal is
called as a regular polygon. A regular polygon is equilateral and equiangular.
Regular Polygon Area
The area of a regular polygon is given by the formula:
Area of regular polygon = 12 * Perimeter * apothem
Where, apothem is a perpendicular that joins the center to the midpoint of any
side of the polygon.
Congruent Polygons
Polygons are congruent when they have same shape and size. Congruent
polygons have the same number of sides, and all corresponding sides and interior
angles are congruent.
Corresponding sides and angles of congruent polygons are congruent.
In the above figure, Polygons ABCDE and FGHIJ are congruent.
Equilateral Polygon
A polygon is an equilateral polygon if all of its sides of the same length. Polygon
sides of an equilateral polygon are equal.
For example, Equilateral triangle, square and rhombus are equilateral polygons.
Equiangular Polygon
A polygon is an equiangular polygon if all of its vertex angles are equal. Regular
polygons are always equiangular.
For example, Equilateral triangle and square are equiangular polygons.
Simple Polygon
A simple polygon is closed polygonal of finitely many line segments that do not
intersect each other; each line segment endpoint is shared by two segments. For
simple polygons, the sum of the exterior angles is always ± 2π.
Inscribed Polygon
A polygon placed inside a circle is inscribed polygon. All the vertices of such
polygon lie on the circumference of the circle.
Circumscribed Polygon
A polygon is circumscribed in a circle, if circle is passes through all the vertices of
the polygon and all polygon sides are included within the circle.
Similar Polygons
Similar polygons are polygons for which all corresponding sides are proportional
and all corresponding angles are congruent. Similar polygon is exactly the same
shape, but can be different sizes.