✅ Degree and Radian Measure Formula ⭐️⭐️⭐️⭐️⭐

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Degrees to Radians

Degrees to radians is a form of conversion used to convert the measurement of angles in geometry. Just like every quantity has a unit of measurement, angles also have units for measurement. For measuring the angles, degrees and radians are used as the units of measurement. It is essential for one to know how to convert degrees to radians since most often, radians are used instead of degrees to measure angles. These are some of the common applications of radian measure: area of a sector of a circle, arc length, and angular velocity. One should essentially use radians when they are dealing with either object moving in circular paths or parts of a circular path. The problem statement can have the angle measured in degrees, but we should always convert the angles in degrees to radians before using them in any calculations. In this article, let’s learn about the degrees to radians formula and the conversion table with solved examples.

Degrees x (π/180) = Radians

Radians  × (180/π) = Degrees

360 Degrees = 2π Radians

180 Degrees = π Radians

What is Degrees to Radians Conversion?

The two units of measurement that are used for measuring angles are degrees and radians. We can convert the measure of any angle given in degrees to radians as per our convenience. This conversion can be done following some very basic calculations. Let us understand the concept of both the units and establish a relationship between them.

Degree:

A degree, which is referred to as the degree of arc or arc degree, is the unit of measuring a plane angle. It is denoted by the symbol (°). 360° is the angle measure for a complete rotation. A complete rotation is denoted by the angle measuring 360° and the instrument used to measure an angle in degrees is known as the protractor.

Radians:

Radian is another unit of measuring an angle in geometry. One radian is the angle that is formed at the center of a circle by an arc whose length is equal to the radius ‘r’ of the circle. One complete counterclockwise rotation is equal to 2π in radians. The below-given image shows the measure of one radian as 57.296°. Also, a right angle is expressed as π/2 radians, and a straight angle is expressed as π radians.

We can compare the measure of angles for a complete rotation in radian and degrees as,

360 Degrees = 2π Radians
180 Degrees = π Radians

Degrees to Radians Formula

Converting degrees to radians formula is a generalized formula used to convert the given value in degrees to value in radians. We can say that to convert degrees into radians, we multiply the angle (in degrees) by π / 180.

Radians = Degrees × π / 180

Derivation of Degrees to Radians Formula

In terms of degrees, one complete counterclockwise revolution is 360° and whereas in radians, one complete counterclockwise revolution is 2π. These statements can be equated as:

One complete counterclockwise revolution in degrees = 360°
One complete counterclockwise revolution in radians = 2π

Degrees to Radians Equation: The relation between degree measure and radian measure can be represented with this equation:

360° = 2π radians.

Hence, 180° = π radians.

We obtain the equation,

1° = (π)/180 radians.

This gives us the formula to make conversions from degrees to radians and vice versa.

Thus, to convert degrees to radians, use this formula = Degree measure × (π/180°). 1° = (π)/180 radians = 0.017453 radians.

To find the value of 1° as explained above, use the π value to be 3.1415926535…

How to Convert Degrees to Radians?

In order to convert degrees to radians manually, we use the formula: Radians = Degrees × (π/180°). We can follow the steps given below to calculate the measure of an angle given in degrees to radians.

  • Note down the measure of the angle given in degrees.
  • We know, 1°= (π)/180 radians. So, to convert the angle given in degrees to radians we multiply it with π/180°.
    Angle in Radians = Angle in Degrees × π/180°.
  • Simplify the values and express the answer in radians.

Example: Convert 90 degrees to radians.

Solution: 90° = 90° × (π/180°) = π/2.

Degrees to Radians Conversion Table

The below-given table shows the radian values for the corresponding angle measures in degrees. The first table shows the degrees to radians angles conversion from 0° to 90°.

Degree30°45°60°90°
Radian0π/6π/4π/3π/2

The second table shows the degrees to radians angles conversion from 120° to 360°.

Degree120º135º150º180º270º360º
Radian(2π)/3(3π)/4(5π)/6π(3π)/2

Degrees to Radians Chart

The below-given chart shows degree measures and their corresponding radian measures. We can also use this chart to convert the degrees to radians to make the calculations easier and faster. In the below-given chart, we can see that 0° equals 0 rad and 360° equals 2π rad

Important Notes on Degrees to Radians

Here is a list of a few points that should be remembered while studying degrees to radians:

  • Angles are measured in degrees and radians.
  • One full revolution is equal to 2π rad (or) 360°.
  • 1° = 0.017453 radians and 1 rad = 57.2958°.
  • To convert an angle from degrees to radians, we multiply it by π/180°.
  • To convert an angle from radians to degrees, we multiply it by 180°/π.

Degrees to Radians Examples

Solution:

We have, ∠AOB = 60°.

Degrees to radians conversion formula is given as (Degrees × π)/180°

Thus converting the given angle we get,

∠AOB in Radians = ∠AOB in degrees × (π/180°)

∠AOB in Radians = 60° × (π/180°)

∠AOB in Radians = π/3

Answer: ∠AOB = π/3 rad.

Example 2: Katie bought a pizza from a nearby store. Her mother gave her a slice that measured (1/4)th of the complete pizza. Katie asked her mother about the central angle that her slice of pizza made. Can you help Katie find the measurement of the angle in degree and radian?

Solution:

Here, the portion Katie got = (1/4)th of the complete pizza. Since the shape of a pizza is circular, and the angle for a complete circle = 360°, thus we can conclude that:

Angle for the complete pizza = 360°

The angle for the (1/4)th slice = 360°/4 = 90°

We know,

The value of this angle in radians = value in degrees × (π/180°)

= 90° × (π/180°) = π/2 rad

Answer: The angle of Katie’s slice in degrees is 90°, and when it is converted from degrees to radians, its measure is π/2 rad.

Example 3: Convert the following degree measure to radian measure.

a) 20 degrees b) 28 degrees

Solution: To convert the following degree measure to radians we will use the following steps:

a) 20° × (π/180°) = π/9 Radians.

b) 28° × (π/180°) = 7π/45 Radians.

FAQs on Degrees to Radians

How Do You Convert Degrees to Radians?

To convert degrees to radians, we have to multiply the given angle (that is in degrees) by π/180°. For this, we use the formula: Radians = Degrees × π/180°.

What Is Degrees To Radians Formula?

Converting degrees to radians formula refers to the formula which converts the given measure of an angle in degrees to its equivalent value in radians. The formula for converting degrees into radians is given as, Radians = Degrees × π / 180°.

How to Convert 30 Degrees to Radians?

We need to first convert the given degrees(30°) in radians using the formula Radians = Degrees × π/180°. Radians = 30° × π/180° = π/6 rad. Hence, 30 degrees can be represented as π/6 rad.

How to Convert 90 Degrees to Radians?

In order to convert 90 degrees to radians, let us multiply the 90° angle by π/180°. Here, we use the formula: Radians = (Degrees × π)/180°. Radians = (90° × π)/180° = π/2 = 0.785 rad. Thus, a right angle is expressed as π/2 radians.

What Are 60 Degrees Converted to Radians?

Let us convert 60 degrees to radians, by multiplying the 60° angle by π/180°. The formula used is: Radians = (Degrees × π)/180°. Radians = (60° × π)/180° = π/3. Hence, 60 degrees converted to radians is π/3.

What Are 180 Degrees in Radians?

We need to first convert the given degrees(180°) in radians using the formula Radians = (Degrees × π)/180°. Radians = (180° × π)/180° = π rad. Hence, a straight angle of 180° can be expressed as π radians.

How Are Radians Used in Everyday Life?

These are some of the ways how radians are used in everyday life:

  • To calculate the area of a sector and the length of an arc.
  • To approximate the length of a chord when the central angle and radius are given.
  • To solve problems involving angular speed.

How to Convert 150 Degrees to Radians?

We need to first convert the given degrees (150°) in radians using the formula Radians = Degrees × π/180°. Radians = 150° × π/180° = 5π/6 rad. Hence, 150 degrees can be represented as 5π/6 rad.

Why Would Radians Be More Useful in the Real World Than Degrees?

Although, for measuring the angles, both degrees and radians are used as the units of measurement, most often, mathematicians prefer using radians instead of degrees to measure angles. Radian measures can be used while calculating the area of a sector of a circle, arc length, and angular velocity. It is essential for one to always use radians when they are dealing with either object moving in circular paths or parts of a circular path. Even if the initial problem statement has the angle measured in degrees, we should always convert these angles to radians before using them in any calculations.

How to Convert 135 Degrees to Radians?

We need to first convert the given degrees (135°) in radians using the formula Radians = Degrees × π/180°. Radians = 135° × π/180° = 3π/4 rad. Hence, 135 degrees can be represented as 3π/4 rad.

How To Use Degrees To Radians Formula?

To convert degrees to radians using degrees to radians formula,

  • Step 1: Check for the value given in degrees
  • Step 2: Multiply the given angle (in degrees) by π/180°. That is, Radians = (Degrees × π)/180°.

What Is the Degree To Radian Formula For Converting a Negative Degree Into Radians?

To convert a negative degree into a radian, the same method is used for positive degrees. Just multiply the given value of the angle in degrees by π/180.

How To Derive Degrees To Radians Conversion Formula?

Since one complete anticlockwise revolution represents the measure in degrees and radians as 360° and 2π, thus 2π = 360°. Also, π = 180°. Thus, the formula is derived as Radians = Degrees × π / 180.

How to Convert 120 Degrees to Radians?

We need to first convert the given degrees (120°) in radians using the formula Radians = Degrees × π/180°. Radians = 120° × π/180° = 2π/3 rad. Hence, 120 degrees can be represented as 2π/3 rad.

How to Convert Degrees to Radians?

The value of 180° is equivalent to π radians. In order to convert any given angle from the measure of degrees to radians, the value needs to be multiplied by π/180. A standard formula for converting from degrees to radians is to simply multiply the number of degrees by π/180

Thus, Angle in Radian = Angle in degree × π / 180° 

Radian Measure Formula

In any circle of radius r, the ratio of the arc length ℓ to the circumference is equivalent to the ratio of the angle θ subtended by the arc at the midpoint and the angle in one rotation. Therefore, radian formula to measure the angles in radians, ℓ2πr=θ2π⟹ ℓ=rθ.

Degree Calculation Formula

The measure of the angle is considered in degrees (°). Radian is often considered while computing the angles of trigonometric functions or periodic functions. Radians are always denoted in terms of pi (π), where the value of pi is equivalent to 22/7 or 3.14.

A degree consists of sub-parts also, stated as minutes and seconds. This conversion is the crucial component of Trigonometry applications. Thus, the Degree formula is: – Radians × 180/π = Degrees

Solved Examples on How to Find Radian Measure From Degree

Example:

Convert 135° into radian measure

Solution:

Given = Angle of 135 degrees

135∘× π/180∘

=3π/4 radians

≈2.35 radians

Example:

Convert 210 degrees to radians.

Solution: Given = Angle of 210 degrees

Angle in radian = Angle in degree x (π/180)

= 210 x (π/180)

= 7π/6

≈3.67

Hence, 210 degrees is equal to 7π/6 (3.67) in radian.

FAQs (Frequently Asked Questions)

1. What is Meant By a Radian?

Answer: In mathematics, the radian is said to be the standard unit of angular measure. An angle’s measurement in radians is numerically equivalent to the length of a concurrent arc of a unit circle. The link or the relationship between the arc length and radius of a circle describes the radian of a circle. Degree formula and radian formula are used to convert, degree to radian or radian to degree.

Radian= Arc Length / Radius Length

Radian=Degree × π / 180

2. What is the Deal of a Circle With Radians Anyway?

Answer: Most of you must be thinking of a circle with respect to degrees: 360° is a complete circle. 180° makes for a half the circle etc… Well, radian measure or radian measure formula is just a distinctive way of talking about the circle or measuring the angle of a circle. Radian measure is just another unit of measure.

Similar to the way we can measure a cricket ground in yards or feet — we can measure a circle in degrees (like the olden days) or in radians (welcome to the big federation!)

Radian also has a close connection to the radius of a circle.

Here are few Degree Measures and their corresponding Radian Measures – 

30° = π/6

45° = π/4

60° = π/3

90° = π/2

120° = 2π/3

135° = 3π/4

150° = 5π/6

180° = π

210° = 7π/6

225° = 5π/5

240° = 4π/3

270° = 3π/2

300° = 5π/3

315° = 7π/4

330° = 11π/6

360° = 2π

How to Convert Degrees to Radians?

The value of 180° is equal to π radians. To convert any given angle from the measure of degrees to radians, the value has to be multiplied by π/180.

Where the value of π = 22/7 or 3.14

The below steps show the conversion of angle in degree measure to radians.

Step 1: Write the numerical value of the measure of an angle given in degrees

Step 2: Now, multiply the numeral value written in step 1 by π/180

Step 3: Simplify the expression by cancelling the common factors of the numerical

Step 4: The result obtained after the simplification will be the angle measure in radians

Let’s have a look at the examples of conversions of different angle measures.

Example 1: Convert 90 degrees to radians.

Solution: Given, 90 degrees is the angle

Angle in radian = Angle in degree x (π/180)

= 90 x (π/180)

= π/2

Hence, 90 degrees is equal to π/2 in radian.

Radians to Degrees Conversion

As we have already discussed, how to convert degrees to radians for any specific angle. Now, let us see how we can convert radians to degrees for any specific angle. The formula to convert radians to degrees is given by:

Radians  × (180/π) = Degrees

Example 2: Convert π/6 into degrees.

Solution: Using the formula,

π/6 × (180/π) = 180/6 = 30 degrees

Radian to Degree Equation

As we know already, one complete revolution, counterclockwise, in an XY plane, will be equal to 2π (in radians) or 360° (in degrees). Therefore, both degree and radian can form an equation, such that:

2π = 360°

Or 

π = 180°

Degrees to Radian Formula

To convert degrees to radian, we can use the same formula as given in the above section.

Degree x π/180 = Radian

Let us see some examples:

Example 3: Convert 15 degrees to radians.

Solution: Using the formula,

15 x π/180 = π/12

Example 4: Convert 330 degrees to radians.

Solution: Using the formula,

330 x π/180 = 11π/6

Negative Degrees to Radian

The method to convert a negative degree into radian is the same as we have done for positive degrees. Multiply the given value of the angle in degrees by π/180.

Suppose, -180 degrees has to be converted into radian, then,

Radian = (π/180) x  (degrees)

Radian = (π/180) x (-180°)

Angle in radian = – π

Degrees to Radians Chart

Let us create the table to convert some of the angles in degree form to radian form.

Angle in DegreesAngle in Radians
0
30°π/6 = 0.524 Rad
45°π/4 = 0.785 Rad
60°π/3 = 1.047 Rad
90°π/2 = 1.571 Rad
120°2π/3 = 2.094 Rad
150°5π/6 = 2.618 Rad
180°π = 3.14 Rad
210°7π/6 = 3.665 Rad
270°3π/2 = 4.713 Rad
360°2π = 6.283 Rad

Conversion Of Some Common Angles

Go through the table given below to understand the conversion of angles in comparison with the amount of turn.

TurnsRadian MeasureDegree Measure
0 turn0 rad
1/24 turnπ/12 rad15°
1/16 turnπ/8 rad22.5°
1/12 turnπ/630°
1/10 turnπ/5 rad36°
⅛ turnπ/4 rad45°
1/2π turn1 rad57.3°
⅙ turnπ/3 rad60°
⅕ turn2π/5 rad72°
¼ turnπ/2 rad90°
⅓ turn2π/3 rad120°
⅖ turn4π/5 rad144°
½ turnπ rad180°
¾ turn3π/2 rad270°
1 turn2π rad360°

Solved Examples

Question 1: Convert 200 degrees into radians.

Solution: By the formula, we know;

Angle in radians = Angle in degree × π/180

Thus,

200 degrees in radians = 200 × π/180 = 10π/9 = 3.491 Rad

Question 2: Convert 450 degrees into radians.

Solution: By the formula, we know;

Angle in radians = Angle in degree × π/180

Thus,

450 degrees in radians = 450 × π/180 = 7.854 Rad

Practice Questions

  1. Transform 700 degrees to radians.
  2. How many radians is equal to 100 degrees?
  3. Convert 550 degrees to radians.

Frequently Asked Questions – FAQs

How do we convert degrees to radians?

To convert the value of the angle in degree, to its equivalent radians, we need to multiply the given value with π/180. For example, the value of 180 degrees is π in radians.

What is the formula to convert radian into degrees?

To convert the value of angle from radian to degrees, we need to multiply the given value with 180/π. For example, the value of 2π:
2π x 180/π = 360 degrees

How to convert 30 degrees to radians?

Multiply 30 degrees to π/180:
30 x (π/180) = π/6
Hence, 30 degrees is equal to π/6 in radians

How pi radians is equal to 180 degrees?

One complete revolution, counterclockwise, in an XY plane, will be equal to 2π (in radians) or 360° (in degrees). Hence, we can write:
2π = 360°
Or
π = 180°
Therefore, pi is equal to 180 degrees

What is the value of 1 degree?

One degree is equal to 0.0174533 radians.

What is the value of 1 radian?

One radian is equal to 57.2958 degrees.

Convert 3 radians to degrees.

3 radian × 180/π = 171.887Deg

Degrees and Radians – Explanation & Examples

Like every other quantity, angles also have units for measurement. Radians and Degrees are two basic units for measuring the angles. There are other units to measure the angles (like gradians and MRADs), but in high school, you will only see these two units.

What are Degrees and Radians?

The most popular unit for measuring angles that most people are familiar with is the degree is written (°). The subunits of a degree are minutes and seconds. There are 360 degrees, 180 degrees for a half-circle (semi-circle), and 90 degrees for a quarter circle (a right triangle) in a full circle or one complete rotation.

Degrees basically state direction and angle size. Facing North means you are facing the direction of 0 degrees. If you turn towards South, you are facing the direction of 90 degrees. If you come back to North after full rotation, you have turned through 360 degrees. Usually, the anticlockwise direction is considered positive. If you turn towards West from North, the angle will be either -90 degrees or +270 degrees.

In geometry, there is another unit for measuring angles, known as the radian (Rad).

Now, why do we need radians when we are already comfortable with angles?

Most computation in mathematics involves numbers. Since degrees are not actually numbers, then the radians measure is preferred and often required to solve problems.

good example that’s similar to this concept is using decimals when we have percentages. Although a percentage can be shown with a number followed by a % sign, we convert it to a decimal (or fraction).

The concept of finding angle by the length of the arc was used long ago. The radian was introduced much later. Roger Cotes gave the concept of radians in 1714, but he didn’t give it this name and just called it a circular measure of an angle.

The term “radians” was first used in 1873. This name, later, got universal attention and got authorization.

In this article, you will learn how to convert degrees to radians and the other way around (radians to degrees). Let’s take a look.

How to convert degrees to radians?

To convert degrees to radians, we multiply the given angle (in degrees) by π/180.

Angle in degrees (°) x π/180 = Angle in radian (Rad)

Where, π = 22/7 or 3.14

Example 1

Convert the following angles from degrees to radians

  1. 30°
  2. 45°
  3. 60°
  4. 90°
  5. 120°
  6. 150°
  7. 180°
  8. 210°
  9. 240°
  10. 360°

Solution

Angle in degrees (°) x π/180 = Angle in radian (Rad)

1. 0° x π/180

= 0 Rad

2. 30° x π/180

= π/6

= 0.5 Rad

3. 45° x π/180

= π/4

= 0.785 Rad

4. 60° x π/180

= π/3

= 1.047 Rad

5. 90° x π/180

= π/2

= 1.571Rad

6. 120° x π/180

= 2π/3

= 2.094 Rad

7. 150° x π/180

= 5π/6

= 2.618 Rad

8. 180° x π/180

= π

= 3.14 Rad

9. 210° x π/180

= 7π/6

= 3.665 Rad

10. 240° x π/180

= 3π/2

= 4.189 Rad

11. 360° x π/180

= 2π

= 6.283 Rad

Example 2

Convert 700 degrees to radians.

Solution

Angle in degrees (°) x π/180 = Angle in radian (Rad)

By substitution,

Angle in radian (Rad) = 700 x π/180.

= 35 π/9

= 12.21 Rad.

Example 3

Convert – 300° to radians.

Solution

Angle in radian = -300° x π/180.

= – 5π/3

= – 5.23 Rad

Example 4

Convert – 270° to radians.

Solution

Angle in radian = -270° x π/180.

= – 3π/2

= -4.71 Rad.

Example 5

Convert 43 degrees, 6 minutes, and 9 seconds to radians.

Solution

First express 43 degrees, 6 minutes, and 9 seconds to degrees only.

43° 6′ 9″ = 43.1025°

43.1025° x π/180 = Angle in radian

= 0.752 Rad.

Example 6

Convert 102° 45′ 54″ to radians.

Solution

102° 45′ 54″ is equal to 102.765°

Angle in radian = 102.765°x π/180.

= 1.793 Rad.

How to Convert Radians to Degrees?

To convert radians to degrees, multiply the radian by 180/ π. So, the formula is given by,

Angle in radian x 180/ π = Angle in degrees.

Example 7

Convert each of the following angles in radians to degrees.

  1. 1.46
  2. 11π/6
  3. π/12
  4. 3.491
  5. 7.854
  6. -8.14
  7. π/180

Solution

Angle in radian x 180/ π = Angle in degrees.

  1. 46 x 180/ π

= 83.69 degrees.

  1. 11π/6 x 180/ π

= 330 degrees.

  1. π/12 x 180/ π

= 15 degrees.

  1. 491 x 180/ π

= 200.1 degrees

  1. 854 x 180/ π

= 450.2 degrees.

  1. -8.14 x 180/ π

= – 466.6 degrees.

  1. π/180 x 180/ π

= 1 degree.

Example 8

Convert the angle π​/5 radians into degrees.

Solution

Angle in radian x 180/ π = Angle in degrees.

By substitution,

π​/5 x 180/ π = 36 degrees.

Example 9

Convert the angle – π​/8 radians into degrees

Solution

​/8 x 180/ π = – 22.5 degrees.

Example 10

The radius of a piece of pizza is 9 cm. If the perimeter of the piece is 36.850 cm, find the angle of the piece of pizza in radians and degrees.

Solution

Let arc length of the piece = x

Perimeter = 9 + 9 + x

36.850 cm = 18 + x

Subtract 18 on both sides.

18.85 = x

So, the arc length of the piece is 18.85 cm.

But, arc length = θr

Where θ = angle in radians and r =radius.

18.85 cm = 9 θ

Divide both sides by 9

θ =2.09 Rad

θ in degrees:

Angle in radian x 180/ π = Angle in degrees.

=2.09 x 180/ π

= 120 degrees.

Example 11

The radius of a sector is 3 m, and its area is 3π/4 m2. Find the central angle of the sector in degrees and radians.

Solution

Given that,

Area of a sector = (r 2θ)/2

Where θ = central angle in radians.

Substitute.

3π/4 = (32 θ)/2

3π/4 = 9θ/2

Cross multiply.

6 π = 36 θ

Divide both sides by 36 to get,

θ = 0.52 Rad.

Convert the angle to degrees.

= 0.52 x 180/ π

= 29.8 degrees.

Example 12

Find the central angle of a sector whose radius is 56 cm and the area is 144 cm2.

Solution

A= (θ/360) πr2

144 = (θ/360) x 3.14 x 56 x 56.

144 = 27.353 θ

Divide both sides by θ.

θ = 5.26

Thus, the central angle is 5.26 degrees.

Example 13

The area of a sector is 625 mm2. If the radius of the sector is 18 mm, find the central angle of the sector in radians.

Solution

Area of a sector = (θr2)/2

625 = 18 x 18 x θ/2

625 = 162 θ

Divide both sides by 162.

θ = 3.86 radians.

The same procedure works just as well for negative angles. Remember that a negative angle is just an angle that is oriented clockwise when drawn in standard position. A negative number of degrees becomes a negative number of radians.

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