How to Find the Radian Measure of an Angle
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.
| 1. | What is Degrees to Radians Conversion? |
| 2. | Degrees to Radians Formula |
| 3. | How to Convert Degrees to Radians? |
| 4. | Degrees to Radians Conversion Table |
| 5. | FAQs on Degrees to 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°.
| Degree | 0° | 30° | 45° | 60° | 90° |
| Radian | 0 | π/6 | π/4 | π/3 | π/2 |
The second table shows the degrees to radians angles conversion from 120° to 360°.
| Degree | 120º | 135º | 150º | 180º | 270º | 360º |
| Radian | (2π)/3 | (3π)/4 | (5π)/6 | π | (3π)/2 | 2π |
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.
☛ Related Articles on Degrees to Radians
Check out the following pages related to degrees to radians:
- Radians to Degree Calculator
- Degrees to Radians Calculator
- Formula For 180 Degree Rotation
- Radians
- Degrees
Degrees to Radians Examples
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Example 1: In a circle with center O, points A and B are two points on the circle. ∠AOB = 60°. Convert ∠AOB's angle measure from degrees to radians.
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.
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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.
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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.
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Degrees to Radians
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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 Find the Radian Measure of an Angle
Source: https://www.cuemath.com/geometry/converting-degrees-to-radians/