Angle an Its Measurement 

An angle is generate by rotating a line segment about any of its end points from some Initial position to some terminal position. The measure of an angle is the amount of rotation.

Important

If the rotation is in anticlockwise sense, the angle measure is positive and if the rotation is in clockwise sense, the angle measured is negative.

Types of Systems of Measuring Angles

A. Sexagesimal System/English Measure or British System 

In Sexagesimal system of measurement, the units of measurement are degrees, minutes and seconds.

Degree- A right angle is divided into 90 equal parts an each part is calle a degree. One degree is denoted by 10.

Minute- A degree is ivided into 60 equal parts and each part is called a minute. One minute is denoted by 1'.

Second- A minute is divided into 60 equal parts and each part is called a second. One secon is denoted by 1".

Important

Sexagesimal system of angles

1 night angle = 90 degree (900)

1 degree = 60 minute (600)

1 minute = 60 second (600)

B. Centesimal System or French System

In centesimal system of measurement, the units of measurement are grades, minutes an seconds.

Grade- A right angle is divided into 100 equal parts and each part is called a grade. One grade is denoted by 1g.

Minute- A grade is divided into 100 equal parts and each part is called a minute. One minute is denote by 1'.

Second- A minute is divied into 100 equal parts and each part is called a second. One secon is enoted by 1".

Important

Centersimal system of angles

1 right angle = 100 grade (1000)

1 grade = 100 minute (100')

1 minute = 100 second (100")

C. Circular Measure 

In circular system of measurement, the unit of measurement is radian.

A radian is the angle subtended at the centre of the circle by an arc equal to the length of the radius of the circle.

Arc Length

The length l of an arc AB of the circle of radius r subtending an angle at the centre of the circle is

So, the length of one full rotation around the circle will subtend angle of 360º Such that 

Important

The radian is a constant angle i.e. it does not depend upon the radius of the circle from which it is derived 

1 Radian =

Problem Solving Trick

Relation Between Degree (D), Grade (G) and Radian (C) 

If D, G and C are respectively the measures of an angle in degrees, grades and radians, then 

The Constant Number

The length of the circumference of a circle always bears a constant ratio to its diameter. Thus the ratio (circumference) : (diameter) is the same for all circles. This constant ratio is always denoted by the Greek letter , so that is a real number.