Measurement of Angle

What is measurement of an angle?
Measurement of an angle is the quantification of the amount of rotation between two intersecting lines, rays, or line segments, at a common point.
The angle is formed at a vertex, when two line segments meet at a point.
The point is called vertex.
In the figure below

1. B is vertex
2. \(\measuredangle ABC\) is an angle
3. AB and CB are arms

System of measurement of Angle

What are the system of measurement of Angle?
There are three main systems used to measure angles: They are

1. Degree system or Sexagesimal system
2. Grade system or Centisimal system
3. Radian system or circular system

What is degree system?
The degree system is a way to measure angles using a unit called a degree (°). In this system, a full circle is divided into 360 equal parts, so one complete turn is 360 degrees.
A straight angle (like a straight line) measures 180°, a right angle is 90°
In this system

1. a right angle = 90°
2. 1°=60' (1 degree=60 minute)
3. 1'=60'' (1 minute=60 seconds)

What is grade system?
The grade system is a way to measure angles using a unit called a degree (\(^g\)). In this system, a full circle is divided into 400 equal parts, so one complete turn is 400 grades. A straight angle (like a straight line) measures \(200^g\), a right angle is \(100^g\)
In this system

1. a right angle = \(100^g\)
2. \(1^g\)=100' (1 grade=100 minute)
3. 1'=100'' (1 minute=100 seconds)

What is radian system?
The radian system is a way of measuring angles based on the properties of a circle. Instead of dividing a circle into 360 degrees, the radian system relates the angle to the arc length of a circle.
In this system, one radian is the angle formed at the center of a circle when the length of the arc is equal to the radius of the circle.

1. a right angle = \(\frac{\pi}{2}\approx 1.57\) radian
2. A straight angle =\(\pi \approx 3.14\) radian
3. a full circle = \(2 \pi \approx 6.28\) radian
4. Angle in radians \(\theta=\frac{l}{r}\)
   Another way of measuring angle in radians is given by
   \(\text{angle}=\frac{\text{arc length}}{\text{radius}}\)
5. So in radians, the angle around a complete circle is
   \(\theta=\frac{l}{r}=\frac{2\pi(r)}{r}=2\pi\)

Slowly drag the slider on the radius. As you do, observe what is happening as things move in arc. As a result of this, how would you define 1 Radian?

What is difference in 60 degree and 1 radian angle?
The length of an arc (the curved part of the circle edge), when equal to radius, defines 1 radian angle, which is \(\approx\) 57.29 degrees.
The length of chord, when equal to radius, which is equal to radius, defines 60 degree angle.

Comparison of Measurement Systems

Angle / Relation	Radian	Degree	Grade
Full circle	\(2\pi\)	360°	400g
Straight angle	\(\pi\)	180°	200g
Right angle	\(\dfrac{\pi}{2}\)	90°	100g
	1 rad\(=\frac{l}{r}\) when \(l=r\)	1°=60′	1g=100′
	—	1′=60″	1′=100″

Drag and Drop Quiz1: Try it

Drag the items from the left and drop them into their correct categories on the right:

360°

2π rad

400^g

π rad

180°

200^g

90°

π/2 rad

Full circle (Degree)

Full circle (Radian)

Full circle (Grade)

Straight angle (Radian)

Straight angle (Degree)

Straight angle (Grade)

Right angle (Degree)

Right angle (Radian)

Score: 0/8

Multiple Choice Quiz2: Try it

Question 1 of 5

Question text will appear here

Quiz Complete! Your Score: 0/5