What is an Angle? Concept of Degree & Radian

The concept of angle is extremely important in mathematics specially if you are solving questions related to geometry and trigonometry. In this chapter we will specifically try to understand the basics of angle and the units used to measure it.

We have tried to explain the concept in as simple way as possible so that any student with weak Math background can grasp the concept. The material below is specifically for Grade 11 students, so you will find some advanced concepts of angle measurement which is only relevant for higher classes.

Concept of Angle

What is an angle?

The angle basically measures the degree of separation between the two lines.

Let us consider the concept of angle in simple manner.
Suppose there is a ray placed in the floor in its initial position.

Now a kid enter into the room and rotate the position of the ray as given in the below illustration. This new position is known as terminal position of the ray

The change in position of ray from initial to final position is measured by using the concept of angle.
In other words, the change in rotation of ray from initial to final position is measured by angle,

Positive Angle and Negative Angle

Positive Angle

If the ray is rotated anti clock wise from the initial position, it is known to form positive angle

In the above figure:
OA ==> Initial position
OB ==> Final/Terminal position
Rotation ==> Anti-clockwise

Hence Angle AOB is positive angle

Negative Angle

If the ray is rotated clockwise from initial position it is known to form negative angle

From the above figure you can observe that:
OA ==> Initial Position
OB ==> Final/Terminal Position
Rotation ==> Clockwise

The angle AOB is negative angle

Rule for Positive and Negative angle

To simplify the above concept, let me state you some rules.

(a) you have to consider horizontal line as 0 degree

(b) If the other ray lies above the horizontal line than positive angle is formed
And if the final ray lies below the horizontal line than negative angle if formed

Angle Measurement in degree

One unit to measure angle between two rays is degree.
Degree is one of the widely used unit of measurement in geometry, so its necessary to understand the concept .

One full revolution

If the ray is completely revolved around to come back at its initial position than the movement is known as complete revolution.

So in one complete revolution; Initial Position = Final Position

In the above figure you can see that the ray OA is revolved such that it come back to its original position.
This 1 complete revolution is measured as 360 degrees

Hence
One Complete Revolution = 360 degree

One Degree

If the rotation of ray is 1/360 of the revolution then the angle is measured to be 1 degree
Hence,
==> 1 degree = 1/360 of revolution

Using the concept above you can find the value of other degree measurements
==> 15 degree = 15/360 of one revolution
==> 30 degree = 30/360 of one revolution
==> 90 degree = 90/360 of one revolution

Degree, Minutes and Seconds

A degree is divided into 60 minutes, and a minute is divided into 60 seconds

1 degree = 60 minutes
1° = 60′,

1 minute = 60 seconds
1′ = 60″

Important Angle Examples

a. 180 degree angle measurement

b. 45 degree and 90 degree angle measurement

c. 270 degree angle measurement
In the below figure you can see angle AOB = 270 degree

If you want to measure same angle using negative angle concept, then the angle AOB will be -90 degree
AOB = -90 degree

Angle measurement in Radians

Apart from degree, the unit of angle measurement is radians.
Radian is also widely used in scientific community, hence its important to understand the concept

What is Radian?

Radian is the angle in a circle form between radius of circle and arc of circle whose length is equal to the radius of circle.

Let us understand the concept with the help of example.

Consider there is a circle of radius r as shown in below illustration.
Then the 1 radian angle formed will be:

You can see in the above illustration that 1 radian angle is formed when radius r of circle form angle with arc of length r
Here,
OA = r
OB = r
arc AB = r
For formation of 1 radian, its necessary to have arc AB of length equal to radius of circle

1.5 Radian

Let us draw diagram of 1.5 radian for our understanding.

In the above illustration it is clear that in order to form 1.5 radian angle, it is necessary to have arc length of 1.5r between the two radius of circle

Angle of complete circle in radian

From the above illustration you can see that the angle formed by complete revolution of circle is 2\pi

Hence,
Angle of complete revolution = 2\pi

Formula for calculation of angle in Radian

If angle is formed with circle of radius r with arc of length l, then the formula for angle calculation is given as

θ = l / r

where,
l is the length of arc
r is the radius of circle
θ is angle formed in radians

Radian and Degree Relationship

We have already found that:
(a) the angle of complete circle in radian is 2\pi
(b) angle of complete revolution in degree is 360

Since both are angles of revolution, thus: (a) = (b)

\ 360=\quad 2\pi \\ \\ 180\quad =\pi \\\ \\ \ we\quad know\quad that\quad { \quad \pi \quad =\frac { 22 }{ 7 } } \\\ \\ 1\quad degree\quad =\quad \frac { \pi }{ 180 } \quad radian\\ \\ 1\quad degree\quad =\quad \frac { 22 }{ 7\quad \times 180 } \\ \\ 1\quad degree\quad =\quad 0.01746\quad radian

From the above calculation we found that:
1 degree = 0.01746 radian

Try to remember this relationship as it would help to solve problems related to trigonometry.

Formula for Degree – Radian conversion

Degree to Radian chart

Questions on Angles

(01) Convert 40° 20′ into radian measure

Given:
Angle = 40 degrees & 20 minutes

First convert the minutes into degrees
40° 20′ ==> 40 + 20/60 degree
40° 20′ ==> \frac { 121 }{ 3 } \quad degree

Now using degree-radian formula
radian\quad =\frac { \pi }{ 180 } \quad \times \quad degree\\\ \\ radian\quad =\frac { \pi }{ 180 } \quad \times \quad \frac { 121 }{ 3 } \\\ \\ radian\quad =\frac { 121\pi }{ 540 }

Hence, radian\quad =\frac { 121\pi }{ 540 } is the answer

(02) Find the radian measurement of – 47°30′

Solution
Angle =negative 47 degree & 30 minutes

First we have to convert the minute part into degree
– 47°30′ ==> – (47 + 30/60)
– 47°30′ ==> – 47.5°

Now using degree-radian formula
radian\quad =\frac { \pi }{ 180 } \quad \times \quad degree\\\ \\ radian\quad =\frac { \pi }{ 180 } \quad \times \quad degree\\\ \\ radian\quad =\frac { \pi }{ 180 } \quad \times \quad -47.5\\\ \\ radian\quad =\frac { -47.5\quad \pi }{ 180 }

(03) Find the radius of the circle in which a central angle of 60° intercepts an arc of length 37.4 cm

Solution
Angle (θ) = 60°
Arc length = 37.4 cm

To find: Radius of circle

First convert the angle from degree to radian using degree radian formula
radian\quad =\frac { \pi }{ 180 } \quad \times \quad degree\\\ \\ radian\quad =\frac { \pi }{ 180 } \quad \times \quad 60\\\ \\ radian\quad =\frac { \pi }{ 3 } \quad \

Here we have to use the following arc formula
θ = l / r

Putting the values
π/3 = 37.4 / r
r = 37.4 *3 / π
r = 37.4 *3 *7/ 22
r= 35.7 cm

Hence radius of circle is 35.7 cm