Angle Measurement

What is Angle?

When a ray OA starting from its initial position OA rotates about its endpoint o and takes final position OB

When angle is formed

Positive and Negative Angles

An angle formed by a rotating ray is said to be positive or negative depending on whether its moves anticlockwise or clockwise direction respectively.

Measurement of Angle

Sexagesimal System / Degree Measure

This is also called an English system.

In this system,

1st right angle = 90o

1o = 60’

1’ = 60’’

Centesimal system of Angle Measurement

This is also known as French system.

\({1}’ = {100}’\\ {1}’ = {100}”\).

Circular system of Angle Measurement

This is very popularly known as radian system.

In this system, the angle is measured in radian

  • A radian is an angle subtended at the centres of a circle by an arc, the whole length is equal to the radius of the circle.


The numbers of radians in an angle subtended by an arc of the circle at the centres is equl to arc/ radius

\(\theta =\frac{arc}{radius}\)

Important Conversions:

  1. \(\pi \,radian=180{}^\circ\)
  2. \(1\,radian=\frac{180}{\pi }\)
  3. \(1{}^\circ =\left( \frac{\pi }{180} \right)\) radian
  4. If D is number of degree, R is the number of radians and G is the number of grade in angle \(\theta\) , then \(\frac{D}{90}=\frac{G}{100}=\frac{2R}{\pi }\)
  5. \(\theta =\frac{1}{r}\) Where \(\theta\)= angle sustended by arc of length 1 at centre of circle .

Practice Problems

Illustration 1

Write (3.25) oin D-M-S

3o– 0.25 X 60’ = 3o15’

Illustration 2

Write in (12.3456)g G- M- S


Illustration 3

Correct in to radian 30o

\(30{}^\circ X\,\frac{\pi }{180}=\frac{\pi }{6}\)

\({{\pi }^{c}}=180\) \({{1}^{c\,}}=\frac{180}{\pi }=\frac{180}{22}X7\,=\,57{}^\circ 16’22”\) \({{1}^{c\,}}\simeq 57{}^\circ\)


Is sin 1 > sin 1o?

Sin 1 = sin 57o

Sin 1o

Hence sin 1 > sin 1o

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