Session 3: Polygons and Curves – Definition, Types, Interactives and Examples

What is a Curve?

A plane figure formed by a continuous and smooth flowing line without retracing any portion of the line other than single points is called a curve. In everyday language, a curve means “not straight”, but in mathematics, a curve can be a straight line as well.

Shown below are some examples of curves.

What is an Open Curve?

An open curve has two endpoints and does not enclose an area within itself. Shown below are some examples of open curves.

What is a Closed Curve?

A closed curve has no endpoints, and it encloses an area. Shown below are some examples of closed curves.

What is a Simple Curve? 

A simple curve does not cross itself at any point. It can be open or closed. Shown below are some examples of simple curves.

What is a Non-Simple Curve?

A curve that crosses its path is called a non-simple curve. It can be open or closed. Shown below are some examples of non-simple curves.

Position of a Point with respect to a Curve

In the curve below, A, B, and C are the three points. 

• The region consisting of all points lying outside a curve is called the exterior of the curve. Point A is an exterior point. 
• The region consisting of all points lying inside a curve is called the interior of the curve. Point B is an interior point. 
• The region consisting of all points which are on the curve is called the boundary of the curve. Point C is on the boundary of the curve.

Polygons

A polygon is a closed curve bounded by several straight lines.

Types of Polygons

A polygon is usually classified by its number of sides, as shown below.

Number of Sides	Name of Polygon	Figure
3	Triangle
4	Quadrilateral
5	Pentagon
6	Hexagon
7	Heptagon
8	Octagon
9	Nonagon (or enneagon)
10	Decagon

Most people prefer to name a polygon with more than ten sides by the number of sides and the suffix “gon.” For example, a 40-sided polygon would be a “40-gon.”

Sides, Vertices and Diagonals of a Polygon

AB, BC, CD and DA are the sides, and A, B, C and D are the vertices of the polygon. 

Line segments AC and BD joining two non-consecutive vertices are called diagonals.

Two sides like AB and AD having a common endpoint are called adjacent sides. 

Two sides like AB and DC having no common endpoint are called non-adjacent sides. 

Angles like ∠A and ∠B having a common arm are called adjacent angles. 

Angles like ∠A and ∠C having no common arm are called non-adjacent angles. 

What is a Convex Polygon?

A diagonal is a line segment that connects any two non-adjacent vertices of a polygon. A polygon is convex if all diagonals remain inside the polygon. Most polygons that you study in geometry will be convex.

What is a Concave Polygon?

If a polygon is not convex, then it is concave (or non-convex). Some diagonals of a concave polygon lie partly or wholly outside the polygon.

What is a Regular Polygon?

A polygon is equilateral if all of its sides are the same length. A polygon is equiangular if all of its angles are the same measure. A polygon is regular if it is both equilateral and equiangular.

Polygons and Curves – Examples

Example 1

In the adjoining figure, name the points that are:

i. in the interior of the polygon

ii. on the exterior of the polygon

iii. on the boundary of the polygon

i. Points L and P are in the interior of the polygon. 

ii. Points J, K, M and Q are on the exterior of the polygon. 

iii. Points A, B, C, D, F, N, and O are on the polygon’s boundary. 

 Remember this!

• A plane figure formed by a continuous and smooth flowing line without retracing any portion of the line other than single points is called a curve.  
• An open curve has two endpoints and does not enclose an area within itself. 
• A closed curve has no endpoints, and it encloses an area. 
• A simple curve does not cross itself at any point. It can be open or closed. 
• A curve that crosses its path is called a non-simple curve. It can be open or closed. 
• A polygon is a closed curve bounded by several straight lines. 
• If two sides have a common endpoint, they are called adjacent sides. 
• If two sides do not have a common endpoint, they are called non-adjacent sides. 
• The angles having a common arm are called adjacent angles. 
• The angles having no common arm are called non-adjacent angles. 
• A diagonal is a line segment that connects any two non-adjacent vertices of a polygon.
• A polygon is convex if all diagonals remain inside the polygon. 
• A polygon is concave if a diagonal lies partly or wholly outside the polygon. 
• A polygon is equilateral if all of its sides are the same length. 
• A polygon is equiangular if all of its angles are the same measure. 
• A polygon is regular if it is both equilateral and equiangular.