Angle Measurement

Angle measurement is an important topic in geometry. In this article, students will learn how to find angle measurements, the different systems for measuring angles, the concept of positive and negative angles, angle conversion, and solved examples.

Angle is a figure formed by two rays, or line segments, sharing a common endpoint.

When a ray OA starting from its initial position OA rotates and takes the final position OB, an angle is formed between OA and OB.

Positive and Negative Angles

An angle formed by a rotating ray is said to be positive if it moves in an anticlockwise direction, and negative if it moves in a clockwise direction.

Measurement of Angle

There are three systems for measuring angles:

1. Degrees

2. Radians

3. Grads

4. Sexagesimal System

2. Centesimal System

1. Circular system
2. Circular System
3. Circular System

Sexagesimal System / Degree Measure

This is also referred to as the English system.

In this system,

1st right angle = 90°

10 = 60

1’ = 60"

Centesimal System of Angle Measurement

This is also known as the French system.

\(\begin{array}{l}{1}^{g} = {100}\\ {1}^{'} = {100}^{''}\end{array}\)

Circular System of Angle Measurement

This is very popularly known as the Radian system.

In this system, the angle is measured in radians

A radian is an angle measured in the ratio of the length of the arc to the radius of the circle.

Note: Please read the instructions carefully before proceeding.

The number of radians in an angle subtended by an arc of the circle at the center is equal to arc/radius.

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

Important Conversions:

1. (\begin{array}{l}180{}^\circ=\pi ,radian\end{array} )

2. $\begin{array}{l}1,radian=\frac{\pi}{180}\end{array}$

3. $1^\circ = \left( \frac{\pi}{180} \right) \ rad$

(\frac{D}{90} = \frac{G}{100} = \frac{2R}{\pi}) , where (D) is the number of degrees, (R) is the number of radians, and (G) is the number of grade in angle (\theta).

θ = 1/r, where r = radius of the circle and θ = angle subtended by arc of length 1 at centre of circle.

6. Some of the standard radian to degree conversions are given below:

30° = π/6 Radian

π/4 Radian = 45°

π/3 Radian = 60°

π/2 Radian = 90°

120° = 2π/3 Radian

3π/4 Radian = 135°

150° = 5π/6 Radian

210° = 7π/6 Radian

225° = 5π/4 Radian

300° = 5π/3 Radian

Related Videos:

Different Types of Angles

Drawing Angles

![Angle Bisector]()

Practice Problems

Illustration 1:

Write 3° 25’ 0" in D-M-S

Solution: 3$^{\circ}$ - 0.25 $\times$ 60’ = 15'

Illustration 2:

Write in 12.3456g G- M- S

Solution: (\begin{array}{l}{12}^{g}-34-56^{”}\end{array})

Illustration 3:

Correct to radians π/6

Solution: $\left(30^\circ \times \frac{\pi}{180}\right) = \frac{\pi}{6}$

| Remember |

\(\pi^c = 180\)

(\begin{array}{l}{{1}^{c,}}=\frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{22}\times 7,=,57^{\circ} 16'22’’\end{array})

‘\(1^{\circ} \simeq 57^{\circ}\)’