An angle is formed by the rotation of a line about one of its endpoints. It is measured by the amount of rotation from the line's initial position to its final position. Common units of angular measurement include degrees (divided into minutes and seconds), grades/centesimal degrees, and radians. A radian is defined as the angle subtended by an arc of a circle equal in length to the circle's radius.

1. Angles
&
units of angular measurements
by
SBR
www.harekrishnahub.com

2. www.harekrishnahub.com
Consider a line 𝑶𝑨. Let the line 𝑶𝑨 rotate about one of its
endpoints, say 𝑶. Let the new position be 𝑶𝑨’. An angle is
said to be generated. 𝑶𝑨 is the initial position and 𝑶𝑨’ is
the final position.
Therefore, an angle is the measure of the amount of
rotation of a line from its initial position to the final position.
An angle is denoted by ∠𝐴𝑂𝐴.
Here, the point of rotation 𝑶 is called the vertex. The initial
and final positions are called the initial and terminal sides
respectively. They form the arms of the angle.

3. www.harekrishnahub.com
If the line 𝑶𝑨 rotates in the anticlockwise direction, we say
that the angle is positive.
If the line 𝑶𝑨 rotates in the clockwise direction, we say that
the angle is negative.

4. www.harekrishnahub.com
Consider a line 𝑶𝑨 rotating about the
origin 𝑶.
Let it start from the position 𝑶𝑿 and
rotate through an angle 𝜽
(anticlockwise) and occupy a position
as shown below.
Then ∠𝑋𝑂𝐴 = 𝜃
The rotating line 𝑶𝑨 is called the
radius vector. The positions 𝑶𝑿 and
𝑶𝑨 are called initial and terminal
positions.
If the rotating line 𝑶𝑨 coincides with
𝑶𝑿, 𝑶𝒀, 𝑶𝑿’ and 𝑶𝒀’, we get 𝟎°, 𝟗𝟎°,
𝟏𝟖𝟎° and 𝟐𝟕𝟎°.
After rotating once, if OA coincides
with the 𝒙 − 𝒂𝒙𝒊𝒔, then 𝜽 = 𝟑𝟔𝟎°.
The angles 𝟎°, 𝟗𝟎°, 𝟏𝟖𝟎°, 𝟐𝟕𝟎° and
𝟑𝟔𝟎° are called quadrant angles.

5. www.harekrishnahub.com
If 𝑶𝑨 lies in 𝑰 quadrant, then we have, 𝟎 ＜ 𝜽 ＜ 𝟗𝟎°
If 𝑶𝑨 lies in II quadrant, then we have, 𝟗𝟎° ＜ 𝜽 ＜ 𝟏𝟖𝟎°
If 𝑶𝑨 lies in 𝑰𝑰𝑰 quadrant, then we have, 𝟏𝟖𝟎° ＜ 𝜽 ＜
𝟐𝟕𝟎°
If 𝑶𝑨 lies in 𝑰𝑽 quadrant, then we have, 𝟐𝟕𝟎° ＜ 𝜽 ＜
𝟑𝟔𝟎°
Let 𝑶𝑨 rotates 𝒏 times (counterclockwise) and comes
back to its position, then the angle is taken as
(𝒏 ✕ 𝟑𝟔𝟎° + 𝜽)

6. www.harekrishnahub.com
Units of Angular
Measurement
1. Sexagesimal measure.
2. Centesimal measure.
3. Radian (or circular)
measure.

7. www.harekrishnahub.com
Sexagesimal measure
• In this units of measurement, an angle is expressed in terms of degrees,
minutes and seconds.
• Consider a horizontal line and a vertical line. The angle between them is a
right angle. The right angle is divided into 90 equal parts. Each part is called
a degree denoted by (⁰). Each degree is in turn divided into 60 equal parts
and each part is called a minute denoted by (‘). Each minute is in turn
divided into 60 equal parts and each part is called a second denoted by (“).
• Therefore, we have 𝟏 𝑹𝒊𝒈𝒉𝒕 𝒂𝒏𝒈𝒍𝒆 = 𝟗𝟎 𝒐
𝟏 𝒐 = 𝟔𝟎′
𝟏′ = 𝟔𝟎"
In this unit, an angle is expressed as: 𝑿 degrees, 𝒀 minutes, 𝒁 seconds.
For example: 𝟕𝟒 𝒐
𝟒𝟖′ 𝟑𝟓"

8. www.harekrishnahub.com
Centesimal measure
• In this system, a right angle is divided into 100 equal parts.
• Each part is called a grade.
• Each grade is further divided into 100 equal parts and each part is called a
minute.
• Each minute is in turn divided into 100 equal parts and each such part is
called a second. 𝟏 𝑹𝒊𝒈𝒉𝒕 𝒂𝒏𝒈𝒍𝒆 = 𝟏𝟎𝟎 𝒈𝒓𝒂𝒅𝒆𝒔 = 𝟏𝟎𝟎 𝒈
𝟏 𝒈
= 𝟏𝟎𝟎′
𝟏′ = 𝟏𝟎𝟎“
𝟗𝟎 𝒐 = 𝟏𝟎𝟎 𝒈
𝟏 𝒐
=
𝟏𝟎𝟎
𝟗𝟎
𝒈
=
𝟏𝟎
𝟗
𝒈
𝟏 𝒈
=
𝟗
𝟏𝟎
𝒐

9. www.harekrishnahub.com
Radian or Circular measure
Consider a circle with centre 𝑶 and radius 𝒓 units. Let 𝑨 and 𝑩 be any points on
the circle, such that the length of the arc 𝑨𝑩 is equal to the radius 𝒓 of the
circle. That is, arc 𝐴𝐵 = 𝑟. Join 𝑶𝑨 and 𝑶𝑩.
Then the angle ∠𝐴𝑂𝐵 will be equal to 𝟏 𝒓𝒂𝒅𝒊𝒂𝒏 or 𝟏 𝒄𝒊𝒓𝒄𝒖𝒍𝒂𝒓 𝒎𝒆𝒂𝒔𝒖𝒓𝒆.
Therefore, a radian is defined as the angle subtended at the centre of the circle
by an arc of length equal to the radius of the circle.