How to Convert Degrees to Radians With Step-by-Step Examples
Radians and degrees are two different ways to measure an angle. To measure the angle in degrees, we need to consider the one complete circle of 360º. Further, by dividing the one complete circle proportionally into 360 parts, each of these parts of the circle is 1º, and for any integer 1 ≤ d ≤ 360º, the measure of the d combined parts obtains dº.
Another system of measurement is by radians, in which angle is measured by measuring the arc length of the unit circle,
Here, we will learn the terms radians and degree in detail.
What are Radians?
The radian (represented by the symbol rad) is the SI unit used to measure angles, and is the angular measure used in the different areas of Mathematics. The unit radian was earlier known as the SI supplementary unit, but now it is considered as an SI derived unit. One radian is defined as the angle subtended at the centre of the circle by an arc whose length is equal to the radius of the circle.
Radians Formula
The angle formed by the half circle is 180º which is equivalent to π radians.
Accordingly, the angle formed by one complete counterclockwise revolution is equivalent to 2π radians.
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Hence, the radians formula in terms of π
1 Radians = 360o/2π
How to Measure Angles in Radians and Degree?
Consider a unit circle given below centred at the origin and let represents an angle formed with the positive x - axis. Then the measurement of an angle in radians is the arc length of the circle turned out by the angle is 0.
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To understand how to calculate this, let us discuss a few examples. If the given angle measures 0 in degree, then the arc length of the circle turned out by an angle is 0. Similarly, If the given angle measures 360 degrees, then the arc length of the circle turned out by an angle is the circumference of a circle with radius equal to 1 or 2π. 1 = 2π . For any other angle measurement in degrees, we need to compute the fraction of the arc length turned out by converting from degrees to radians, as shown in the example given below.
Example
Find the Value of the Right Angle in Radians.
Solution:
As the value of right angle is 90º in degrees
Accordingly, the value of 90º in radians = 90ºx π/180 = π/2
Therefore, the value of right angle or 90º in radians is equal to π.
Radius and Degrees Formula
Radius and degrees formulas given below helps us to find the angles given in degrees to radians, and vice versa.
To convert an angle given in degrees to radians , simply multiply the angle by π/180
To convert an angle given in radians to degrees , simply multiply the angle by 180/π.
Another simple formula to convert degrees to radians is :
\[\frac{D}{90}=\frac{R}{\frac{\pi }{2}}\]
The above formula to convert degrees to radians is true because the number of radians in 90º is equal to π/2 radians. Hence, we can convert the angle given in degree to radians using the above formula also.
Degrees and Radians Differences
A degree is a unit of measurement that is used to measure spheres, circles and angles whereas radians is a unit of measurement that is only used to measure angles.
A degree is a unit of measurement based on the amount of rotation or turn while a radian is a unit of measurement based on the arc length obtained by each angle.
Degrees are further classified into minutes and arc seconds whereas radians have no subdivision but use decimals for smaller and fractional angles.
A degree is a unit used to measure angle which equals 1/360 of one complete circle whereas radians is the angle subtended by a circular arc which has a similar length as its radius. Radians states that the circle follows 360 or 2π radians.
Degrees are most commonly used in simple Mathematics whereas radians are used in advanced Mathematics.
How to Use a Radian Calculator?
Following are the steps that should be followed to use a radian calculator.
Step 1: Enter the angle in radians in the input column.
Step 2: Click on convert to degrees button
Step 3: As you click on the button, the conversion value from radian to degree will be displayed in the output column.
How to Use a Degree to a Radian Calculator?
Following are the steps that should be followed to use a degree to radians calculator.
Step 1: Enter the angle in degrees in the input column
Step 2: Click on convert to radians button
Step 3: As you click on the button, the conversion value from degree to radians will be displayed in the output column.
Degrees to Radians Conversion Values Table
Fun Facts
In one whole circle, there are 2π radians, which implies that half a circle (180 degrees) is equal to π radians.
1 Radians = 180/π or 57.2958 degrees.
Radians and Degrees Examples With Solutions
1. Convert the Following From Radians to Degrees.
π radians
π/2 radians
Solution:
To convert the given values in radians to degrees, we use the following radians to degree formula:
Radians = 180/π (Angle in Degrees)
1. π radians = π x 180/π = 180º
2. π/2 radians = π/2 x 180/π = 90º
2. Convert the Following From Radians to Degrees
60º
180º
Solution:
To convert the given values in degrees to radians, we use the following radians to degree formula:
Degrees = π/180 (Angle in Radians)
1. 60o x π/180 =π/3
2. 180o x π/180 = π
FAQs on Radians: Meaning, Formula & Key Examples
1. What is a radian and why is it used in mathematics?
A radian is the standard SI unit for measuring angles, based on the fundamental properties of a circle. It is widely used in higher mathematics, especially trigonometry and calculus, because it simplifies complex formulas by directly relating an angle to the arc length and radius of a circle, making it a more natural unit than degrees.
2. How is a radian defined in relation to a circle?
One radian is formally defined as the angle created at the center of a circle by an arc whose length is exactly equal to the circle's radius. This means if you take the radius of a circle and lay it along its circumference, the angle formed at the center is 1 radian.
3. What is the fundamental formula to convert degrees to radians and vice versa?
The conversion between degrees and radians is based on the core relationship that a half-circle is 180° or π radians.
- To convert from degrees to radians, you multiply the angle in degrees by the factor (π / 180).
- To convert from radians to degrees, you multiply the angle in radians by the factor (180 / π).
4. How many radians are there in a full circle, and what is the approximate value of 1 radian in degrees?
A full 360° circle contains exactly 2π radians. Based on the conversion formula (180/π), the value of 1 radian is approximately 57.2958 degrees. This is a constant value derived directly from the properties of a circle.
5. Why are radians, especially expressions involving π, preferred over degrees in higher-level mathematics like calculus?
Radians are preferred in calculus and other advanced fields because they are 'dimensionless' and lead to simpler, more elegant formulas. For example, the derivative of sin(x) is simply cos(x) when x is in radians. If x were in degrees, the derivative would be a more cumbersome (π/180)cos(x), complicating many calculations in physics and engineering.
6. Can you provide examples of converting common angles like 30°, 45°, 60°, and 90° into radians?
Yes, by applying the conversion formula (multiplying by π/180), we can find the radian equivalents for these common angles:
- 30° in radians = 30 × (π/180) = π/6 rad
- 45° in radians = 45 × (π/180) = π/4 rad
- 60° in radians = 60 × (π/180) = π/3 rad
- 90° in radians = 90 × (π/180) = π/2 rad
7. How are radians different from degrees as units of angle measurement?
The key difference is their origin. Degrees are an arbitrary human-made unit, dividing a circle into 360 parts for historical and divisional convenience. In contrast, radians are a natural mathematical unit derived directly from a circle's own properties (its radius). This makes radians fundamentally linked to geometry and essential for formulas describing natural phenomena.
8. How do you convert an angle from a format of degrees, minutes, and seconds (DMS) into radians?
To convert an angle from the DMS format into radians, you must first express the entire angle in decimal degrees. The process is as follows:
- Convert minutes to degrees by dividing the number of minutes by 60.
- Convert seconds to degrees by dividing the number of seconds by 3600.
- Add the decimal values from the minutes and seconds to the main degree value.
- Multiply this final decimal degree value by (π/180) to get the equivalent angle in radians.
9. What is the relationship between arc length, radius, and the central angle measured in radians?
The relationship is expressed by the simple and powerful formula: θ = s / r. In this formula:
- θ (theta) is the central angle measured in radians.
- s represents the length of the arc.
- r represents the radius of the circle.
10. In which real-world fields or applications are radians most commonly used?
Radians are essential in many scientific and technical fields beyond the classroom. Key applications include:
- Physics: For calculating angular velocity, rotational motion, and wave oscillations.
- Engineering: In the design of circular objects like gears, pipes, and in electrical engineering for analysing AC circuits.
- Computer Science: For programming rotations in computer graphics, animation, and game development.
- Astronomy: To measure angular distances and sizes of celestial objects.
