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)**

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