Introduction
Most of the shapes that you are familiar with and that you see every day are polygons.
The word itself means 'many angles', and every polygon has the same number of angles as sides.
Most of the shapes that you are familiar with and that you see every day are polygons.
The word itself means 'many angles', and every polygon has the same number of angles as sides.
What is a polygon?
Distinguishing polygons from other shapes by eye is relatively simple.
You probably understand the distinction between polygons and other shapes just from looking at the diagrams above. In precise terms, though, the definition of a polygon is:

Look at the diagrams below and then answer the questions.
Polygons can be classified according to the number of sides they have. Look at the table below and use it to answer the questions that follow.

Describing polygons
The name of a polygon indicates only the number of sides it has, and there are many different shapes that have the same
number of sides. So to describe a polygon accurately we need to look at other aspects of its shape.
To start with, it is useful to label the vertices and sides. The vertices of a shape are often named in sequence either
clockwise or anticlockwise with letters from the alphabet.To help us reason about
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle and line relationships in polygons, we often talk about sides, angles or vertices being adjacent (next to one another). So in Fig.4 above: side AB is adjacent to side BC
 vertex D is adjacent to vertex E
 angle x is adjacent to angle y
Two adjacent sides of a polygon create an interior angle. In Fig.5 below, the interior angles of three polygons have been marked.
If all of a polygon's interior angles are less than 180, it is called a convex polygon. If one or more of its angles are
reflex
A reflex angle is over 180 but less than 360.reflex angles (over 180), it is described as concave.We also refer to the exterior angles of a convex polygon. These are the angles formed by a side and an adjacent side that has been extended, as illustrated below.
We can
produce
When a line segment is produced it is extended in the same direction.produce (extend) each of the other sides of the above irregular
pentagon
A pentagon is a fivesided polygon.pentagon. Press the button to see this. Fill in the values of the exterior and interior values at each
vertex
vertex in the table below.
For each vertex, add together the interior and the exterior angles.

What you should be able to see is that the interior and exterior angles at every vertex of this polygon sum to 180.
What is a rectangle?
You will already recognize what a rectangle looks like, but how would you describe it to someone who does not recognize
the name you use for it?Click on the figure below to interact with the model.
You could start by describing how many sides it has.
We have already established that there are plenty of shapes with the same number of sides as a rectangle that are not rectangular, so we need to supply more information about it to define it adequately. We know that:
'Every rectangle has two sets of
parallel
Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.parallel sides.'We also know that:
'The four interior angles of a rectangle all measure 90.'
So we have defined a rectangle. In fact, we don't need to mention the existence of its parallel sides, since any quadrilateral with four interior angles of 90 will have two sets of parallel sides. So our definition can be shortened:

Interestingly, a square is actually a special type of rectangle: it is a rectangle with four sides of equal length.
Click on the figure below to interact with the model.
What is a parallelogram?
A
parallelogram
A parallelogram is a quadrilateral with two sets of parallel sides.parallelogram is a quadrilateral that has two sets of parallel sides. In fact, this is all that is need to define a parallelogram. Parallelograms
can often be found in mechanical designs, such as in Fig.10.Adjust the shape of the parallelogram below by dragging its corners and sides. Notice how its angles are related.
Click on the figure below to interact with the model.
Opposite sides in a parallelogram are always parallel, which ensures that there are two pairs of sides made up of lines of the same length. Drag on one side of the parallelogram in Fig.11 to alter the internal angles.
Since a parallelogram is not defined in terms of the size of its angles, it may have four angles of 90. A
rectangle
A rectangle is a quadrilateral with four interior angles of 90.rectangle is, therefore, a type of parallelogram.And just as a square is a rectangle with equallength sides, so a rhombus is a parallelogram with equallength sides.
Click on the figure below to interact with the model.
Click on the figure below to interact with the model.
What is a trapezium?
A
trapezium
A trapezium is a quadrilateral with one set of parallel sides and one set of nonparallel sides.trapezium is a quadrilateral with one set of parallel lines and one set of nonparallel lines. All the shapes in Fig.14 below are trapezia.Investigate the properties of trapezia with the shape below in Fig.15.
Click on the figure below to interact with the model.
Regular polygons
Regular polygons have equal sides and equal interior angles. All the shapes below are regular polygons.
Fig.17 below shows a
regular polygon
A regular polygon has sides of equal length and interior angles of the same size.regular polygon. Connect each vertex to the centre by clicking on the button next to the shape.Click on the figure below to interact with the model.
So a regular polygon with n sides can be thought of as a pattern of n congruent isosceles triangles.
Click on the figure below to interact with the model.
Summary
Polygons are closed figures bounded by straight line segments.
Polygons are often classified according to how many sides they have (e.g. pentagons, hexagons, heptagons).
Every polygon is either concave or convex.
For every interior angle in a convex polygon, there is a corresponding exterior angle.
Regular polygons have sides of equal length and interior angles of equal size.
Polygons are closed figures bounded by straight line segments.
Polygons are often classified according to how many sides they have (e.g. pentagons, hexagons, heptagons).
Every polygon is either concave or convex.
For every interior angle in a convex polygon, there is a corresponding exterior angle.
Regular polygons have sides of equal length and interior angles of equal size.
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