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# Angle Measurement

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

## What is Angle?

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

There are three systems for measuring angles:

1. Sexagesimal system
2. Centesimal system
3. Circular system

### 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.

$$\begin{array}{l}{1}^{g} = {100}’\\ {1}’ = {100}”\end{array}$$
.

### 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 equal to arc/ radius

$$\begin{array}{l}\theta =\frac{arc}{radius}\end{array}$$

Important Conversions:

1. $$\begin{array}{l}\pi \,radian=180{}^\circ\end{array}$$
2. $$\begin{array}{l}1\,radian=\frac{180}{\pi }\end{array}$$
3. $$\begin{array}{l}1{}^\circ =\left( \frac{\pi }{180} \right)\end{array}$$
4. If D is number of degree, R is the number of radians and G is the number of grade in angle
$$\begin{array}{l}\theta\end{array}$$
, then
$$\begin{array}{l}\frac{D}{90}=\frac{G}{100}=\frac{2R}{\pi }\end{array}$$
5. $$\begin{array}{l}\theta =\frac{1}{r}\end{array}$$
Where
$$\begin{array}{l}\theta\end{array}$$
= angle subtended by arc of length 1 at centre of circle .
6.  Some of the standard radian to degree conversions are given below:
• π/6 Radian = 30°
• π/4 Radian = 45°
• π/3 Radian = 60°
• π/2 Radian = 90°
• 2π/3 Radian = 120°
• 3π/4 Radian = 135°
• 5π/6 Radian = 150°
• 7π/6 Radian = 210°
• 5π/4 Radian = 225°
• 5π/3 Radian = 300°

### Related Videos:

Different Types of Angles

Draw Angles

### Practice Problems

Illustration 1:

Write (3.25) oin D-M-S

Solution: 3o– 0.25 X 60’ = 3o15’

Illustration 2:

Write in (12.3456)g G- M- S

Solution:

$$\begin{array}{l}{{12}^{g}}-34′-56”\end{array}$$

Illustration 3:

Correct in to radian 30o

Solution:

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

Illustration 4:

If the angles of a triangle are in the ratio 1 : 2 : 3, then find the angles in degrees.

Solution:

Let the angles be x, 2x and 3x.
Then, x + 2x + 3x = 180°    ….[∵ sum of the angles of a triangle = 180°] ∴ 6x = 180°
∴ x = 30°, 2x = 60° and 3x = 90°

Illustration 5:

Which of the following pairs of angles are not coterminal?
(A) 330°, − 60°           (B) 405°, − 675°
(C) 1230°, − 930°      (D) 450°, − 630°

Solution:

Here, 405° − (− 675°) = 1080° = 3(360°),
1230° − (− 930°) = 2160° = 6(360°)
and 450° − (− 630°) = 1080° = 3(360°) are a multiple of 360°. Hence, these angles are co-terminal.
Now, 330° − (− 60°) = 390° which is not a multiple of 360°. So, these pair of angles are not co-terminal.

Illustration 6:

If the arcs of the same length of two circles subtend 75° and 140° at the centre, then the ratio of the radii of the circles is _____.

Solution:

S1 = S2
∴ r1 × θ1 = r2 × θ2
∴ r1 [75π / 180] = r2 [140 π / 180] ∴ r1 / r2 = 140 / 75
∴ r1 / r2 = 28 / 15

Illustration 7:

The perimeter of a sector of a circle, of area 36π sq.cm., is 28 cm. What is the area of the sector?

Solution:

Area of circle = πr2 = 36π sq.cm
∴ r = 6 cm
Now, perimeter of sector = 2r + S
But, perimeter is given to be 28 cm.
∴ 28 = 12 + S
∴ S = 16 cm
Area of sector = 1 / 2 × r × S = 1 / 2 × 6 × 16 = 48 sq.cm

## Frequently Asked Questions

### What do you mean by radian?

A radian is defined as the ratio of the length of the arc that the angle subtends to a circle, divided by the length of the radius of the same circle.

### What is the best way of measuring an angle?

The best way to measure an angle is to use a tool named protractor.

π radian = 1800.

### How to convert an angle given in degrees to radians?

To convert angle in degrees to radians, we have to multiply the given value with π/180.

Test your Knowledge on Angle Measurement