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Angles: An Introduction

An angle is formed when two rays are joined together at a common point. The common point here is called node or vertex and the two rays are called  arms of the angle. The angle is represented by the symbol ‘∠’. The word angle came from the Latin word “Angulus”. Learn more about lines and angles here. The angle is usually measured in degrees, using a protractor. Degrees 30°, 45°, 60°, 90°, 180° shows different angles here. The types of angles are based on the values of angles in degrees.  We can also represent angles in radians, i.e., in terms of pi (π). 180 degrees is equal to π in radians. 

Definition Of Angles

An angle is a form of geometrical shape, that is constructed by joining two rays to each other at their end-points. The angle can also be represented by three letters of the shape that define the angle, with the middle letter being where the angle actually is (i.e.its vertex). Angles are generally represented by Greek letters such as θ, α, β, etc.


Adjacent Angles Diagram

Eg. ∠ABC, where B is the given angle. Angle measurement terms are – degree °, radians or gradians. The amount of rotation about the point of intersection of two planes (or lines) which is required to bring one in correspondence with the other is called an Angle.

Types of Angles and Their Properties

There are six types of angles. Each type of angle has a unique identification on the basis of angle measurement. Let us read about each type of angle individually along with their properties.

Types of Angles

Angle Based on Rotation

Based on the direction of measurement or the direction of rotation, angles can be of two types: • Positive Angles • Negative Angles

Positive Angles:


An angle measured in the counterclockwise (anti-clockwise) direction is a positive angle. In other words, positive angles are those angles that are rotated from the base in the anti-clockwise direction.


Negative Angles:


Negative angles are those angles that are measured in a clockwise direction from the base. In other words, negative angles are those angles that are angles are rotated from the base in the clockwise direction.

Positive and Negative Angles

Parts of Angles

Parts of Angles

Angle Based on Rotation

To measure everything in this world, we need a unit in a similar angle measurement requires three units of measurement :

Degree of an Angle:

It is represented by ° (read as a degree). It most likely comes from Babylonians, who used a base 60 (Sexagesimal) number system. In their calendar, there was a total of 360 days. Hence, they adopted a full angle to be 360°. First, they tried to divide a full angle into angles using the angle of an equilateral triangle. Later, following their number system (base 60), they divided 60° by 60 and defined that as 1°. Sometimes, it is also referred to as arc degree or arc-degree which means the degree of an arc.
An angle is said to be equal to 1° if the rotation from the initial to the terminal side is equal to 1/360 of the full rotation. A degree is further divided into minutes and seconds. 1′ (1 minute) is defined as one-sixtieth of a degree and 1” (1 second) is defined as one-sixtieth of a minute. Thus, 1°= 60′ = 3600”

Degree Measurement

Radian of an Angle:

This is the SI unit of angle. Radian is mostly used in Calculus. All the formula for derivatives and integrals hold true only when angles are measured in terms of a radian. It is denoted by ‘rad’. The length of the arc of a unit circle is numerically equal to the measurement in radian of the angle that it subtends. In a complete circle, there are 2π radians. 360 = 2π; radian Therefore, 1 radian = 180°/π

Gradian of an Angle:

This unit is least used in Maths. It is also called a gon or a grade. An angle is equal to 1 gradian if the rotation from the initial to terminal side is 1/400 of the full rotation. Hence, the full angle is equal to 400 gradians. It is denoted by ‘grad’.

gradian of an angle

Variable Expressions

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