# 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.

Note:

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

${{12}^{g}}-34′-56”$

Illustration 3

$30{}^\circ X\,\frac{\pi }{180}=\frac{\pi }{6}$
 Remember ${{\pi }^{c}}=180$ ${{1}^{c\,}}=\frac{180}{\pi }=\frac{180}{22}X7\,=\,57{}^\circ 16’22”$ ${{1}^{c\,}}\simeq 57{}^\circ$