Basic Geometrical Ideas

Polygons and Angles

Learning about geometric shapes and figures is very much important in order to build a relationship with the various structures present around us. Have you heard about a polygon? What exactly do you infer? From real-life objects, a STOP sign or a starfish, both are forms of a polygon. In this lesson, we’ll learn what is an angle and what is a polygon.

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Definition of Polygon

In simple mathematics, a polygon can be any 2-dimensional shape that is formed with straight lines. Be it quadrilaterals, triangles and pentagons, these are all perfect examples of polygons. The interesting aspect is that the name of a polygon highlights the number of sides it possesses.

For example, a triangle has three sides, and a quadrilateral has four sides. So, any shape that can be drawn by connecting three straight lines is called a triangle, and any shape that can be drawn by connecting four straight lines is called a quadrilateral.

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Types of Polygons

It should be known that polygons are categorized as different types depending on the number of sides together with the extent of the angles. Some of the prime categories of polygons include regular polygons, irregular polygons, concave polygons, convex polygons, quadrilateral polygons, pentagon polygons and so on.

Some of the most well-known polygons are triangles, squares, rectangles, parallelograms, pentagons, rhombuses, hexagons etc.

Regular polygon

Considering a regular polygon, it is noted that all sides of the polygon tend to be equal. Furthermore, all the interior angles remain equivalent.

Irregular polygon

These are those polygons that aren’t regular. Be it the sides or the angles, nothing is equal as compared to a regular polygon.

Concave polygon

A concave polygon is that under which at least one angle is recorded more than 180 degrees. Also, the vertices of a concave polygon are both inwards and outwards.

Convex polygon

The measure of interior angle stays less than 180 degrees for a convex polygon. Such a polygon is known to be the exact opposite of a concave polygon. Moreover, the vertices associated to a convex polygon are always outwards.

Quadrilateral polygon

Four-sided polygon or quadrilateral polygon is quite common. There are different versions of a quadrilateral polygon such as square, parallelogram and rectangle.

Pentagon polygon

Pentagon polygons are six-sided polygons. It is important to note that, the five sides of the polygon stay equal in length. A regular pentagon is a prime type of pentagon polygon.

Formulae Related to Polygon

(N = count of sides and S = distance from center to a corner)

Regular polygon Area = (1/2) N sin(360°/N) S2

The number of diagonals = 1/2 N(N-3)

Summation of the interior angles of the polygon = (N – 2) x 180°

The count of triangles (while drawing all the diagonals through a single vertex) in a polygon = (N – 2)

What Is An Angle?

The study of angles is very important whenever we are trying to understand polygons and their properties. To be precise, when two rays hold a common endpoint, in this case, the two rays together form an angle. Therefore, an angle is formed by two rays initiating from a shared endpoint. These two rays creating it are termed as the sides or arms of the angle. For representing an angle the symbol “∠” is used in geometry.

Angles of Polygons

One must keep in mind that all polygons possess internal angles and external angles. In addition, a polygon’s external angle can be termed as that which is extended on one side. Here are certain rules which are followed regarding angles of a polygon.

  • Exterior Angle of a Polygon: All the Exterior Angles associated to a polygon add to form a sum 360°.
  • Interior Angle of a Polygon: The Interior and exterior angle are evaluated through the same line, therefore, they add up to 180°.

That is, Interior Angle = 180° − Exterior Angle

Question For You

Q. What is the exact interior angle for a regular octagon?

Ans: Since a regular octagon possesses 8 sides.

Therefore, Exterior Angle = Total Degree of Polygon/Side= 360° / 8 = 45°

Hence, Interior Angle = 180° − 45° = 135°

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