How to Convert Degrees to Radians?
Degrees and radians are two different units that are used for the measurement of the angles. The conversion of degrees to radians is considered while measuring the angles in Geometry. The measure of the angle is generally denoted by degrees, having the symbol °. An angle can be determined by two different kinds of units, that are, degrees and radians. You can convert one form of the representation of any mathematical angle to the other by using simple formulas. A degree also has its sub-parts that are minutes and seconds. This conversion plays a major part in the trigonometry applications. In this article, we will learn about how to convert degrees to radians, degrees to radians formula, and look at some solved examples based on how to convert degrees to radians formula. Let us first look at the degree to radian conversion.
The Degree to Radian Conversion
The value of 180° equals \[\pi\] radians. For converting any given angle from the measure of its degrees to the radians, you need to multiply the value by \[\frac{\pi}{180}\].
The value of \[\pi\] is \[\frac{22}{7}\] or 3.14.
Degrees to Radians Formula
We already learned about how to change degrees to radians for any given angle. Let us learn about how to convert degrees to radians formula. The formula to change the degree to radian is given as follows:
Degree \[\times \frac{\pi}{180}\] = radians
How to convert Degrees to Radians
Let us now look at a step by step procedure for converting degrees to radians.
1. Jot down the degrees that you want to convert into radians. Consider the following examples:
Example 1: 120°
Example 2: 30°
Example 3: 225°
2. Then multiply the degrees by \[\frac{\pi}{180}\].
Example 1: \[120 \times \frac{\pi}{180}\]
Example 2: \[30 \times \frac{\pi}{180}\]
Example 3: \[225 \times \frac{\pi}{180}\]
3. Then simply carry out the multiplication by multiplying the degrees by π/180. Think of it as if you are multiplying two fractions. The first fraction consists of the degrees in the numerator and 1 as the denominator, and the second fraction consists of π in the numerator and has 180 in the denominator.
Example 1:
\[120 \times \frac{\pi}{180}\]
= \[\frac{120\pi}{180}\]
Example 2:
\[30 \times \frac{\pi}{180}\]
=\[\frac{30\pi}{180}\]
Example 3:
\[225 \times \frac{\pi}{180}\]
= \[\frac{225\pi}{180}\]
4. The last step is to simplify. Now, you have to put each fraction in its lowest terms to get the final answer. Find the largest number which can evenly divide to the numerator and the denominator of each fraction and use it for simplifying each fraction.
Example 1:
\[120 \times \frac{\pi}{180}\]
= \[\frac{120\pi}{180} ÷ \frac{60}{60} \]
= \[ \frac{2}{3 \pi} \] radians
Example 2:
\[30 \times \frac{\pi}{180}\]
= \[\frac{30\pi}{180} ÷ \frac{30}{30} \]
= \[\frac{1}{6\pi}\] radians
Example 3:
\[225 \times \frac{\pi}{180}\]
= \[\frac{225\pi}{180} ÷ \frac{45}{45} \]
= \[\frac{5}{4π}\] radians
This is a very simple method and you can easily convert degrees to radians with this simple procedure. Let us look at how to convert angles to radians in detail.
How to Convert Angles to Radians
You learned about how to change degrees to radians. Now let us learn about how to convert the angle to radians.
The angle that is made when the radius is wrapped around the circle is given as follows:
1 radian is approximately 57.2958°.
Consider a half-circle which has radians, which equals to 180°.
Hence, 1 radian = \[ \frac{180^{0}}{\pi} \] = 57.2958°
If you want to convert a degree or an angle to radians, simply multiply the angle by and then divide it by 180.
Take a look at the table below of the angles and their conversion to radians.
To convert degree measure into radian measure, students may directly use the formula. Multiply the given value in degrees with \[\frac{\pi}{180}\]. This is a simple step and students can use this to find radian measure. However, the table given above tells the radian and approximate radian values of the most common angles. Students may use this table for easier and faster calculations. For example, if a student wants to calculate the radian measure of 30°, 60° and 90°, he or she may refer to the table. The radian measures of the following measures in degree will be \[\frac{\pi}{6}\], \[\frac{\pi}{3}\] and \[\frac{\pi}{2}\] and the radian values will be 0.524, 1.047 and 1.571 respectively.
Uses of Radian
Radian is another unit to measure angles and also the SI unit of measurement of angles. It is defined as the angle made at the center of the circle by an arc which is equal to the length of the radius of the circle.
It is denoted by ‘rad’ or by using the letter c. Angle written without a unit means it is written in radian measure. Some examples are 4 rad, \[\frac{\pi}{2}\] or 90°
Calculus and other fields of mathematics use radians as the unit of measurement. It is also used in the areas of science.
Solved Examples
Let us now look at some of the solved examples for your better understanding of how to convert degrees to radians and radians to degrees.
Example 1
Convert 120° to radians.
Solution:
To convert 120° to radians, consider the formula
\[\text{Angle in radians} = \text{angle in degrees} \times (\frac{\pi}{180})\]
Therefore, \[120° \times (\frac{\pi}{180}) \]
= \[(\frac{2\pi}{3})\] radians ≈ 2.09 radians
Example 2
Convert 1.4 radians to degrees.
Solution:
To convert radians to degrees, consider the following formula:
\[\text{Degree} = \text{radians} \times \frac{180}{\pi} \]
= \[1.4 \times 180 = 252 \]
= \[1.4 \times \frac{180}{\pi} = \frac{252}{\pi} \]
≈ 80.2°
Example 3
Convert \[\frac{4\pi}{9} \] radians to degrees.
Solution:
To convert radians to degrees, students may use the following formula:
\[\text{Degree} = \text{radians} \times \frac{180}{\pi} \]
= \[\frac{4\pi}{180} \times \frac {180}{\pi} \]
= \[\frac{4\pi}{180} \times \frac {180}{\pi} \]
= \[4\pi \times \frac {180}{\pi} \times \pi\]
= \[4 \times 20\]
= 80°
Practice Problems
1. Convert into radian measure: 700⁰.
Ans: 12.21 Rad.
2. Determine the radians equivalent to 450 degrees.
Ans: 7.854 Rad.
3. Convert into radian measure -300°.
Ans: -5.23 Rad.
Summary
We have discussed above the two ways to measure the angle of a circle: in degrees and radians. We may convert between the two using formulae. Two rad or 360° are the same as one full circle. One radian (rad) equals 57.2958 degrees, and one degree is equal to 0.017453 radians (rad). An angle is multiplied by π/180⁰ to convert it from degrees to radians. This article has also covered how to convert 60° into radians. Try to solve the practice problems provided in this article using the degrees into radians conversion formula.
FAQs on Degrees to Radians
1. How to convert 1 Degree to Radians?
There is a simple formula for converting radians to degrees.
1π radian = 180°.
Therefore you can convert one unit of measure to the other easily.
One degree is equal to 0.01745329252 radians.
Therefore,
\[1° = \frac{\pi}{180°}\]
= 0.005555556π
= 0.01745329252 rad
2. What is a Radian?
The measure of a given angle can be determined by the amount of the rotation that takes place from the initial side to the terminal side. One way in which you can measure angles is in radians. To define a radian, use the central angle of a circle. Consider an angle whose vertex is the center of the circle. One radian here refers to the measure of the central angle which intercepts the arc s that is equal in the length to the radius r of the given circle.
As you know that the circumference of a circle is given by 2πr one revolution around the given circle having a radius r corresponds to the angle of 2π radians since
\[\frac{s}{r} = (\frac{2πr}{r}) \]
= 2π
3. What is π ?
π or pi is one of the important mathematical constants. It describes the ratio of a circle's circumference to its diameter. The circumference is the distance around the circle whereas the diameter (longest chord of a circle) is the line passing through the centre that touches two points on the boundary of the circle. The value of pi is an irrational number which means that the number of places after the decimal is infinite and we cannot really count it. The value of pi (π) is equal to 3.14159 (approximately). For easier calculations, students use 3.14 or \[\frac{22}{7}\] as the value of pi.
4. How to convert 24 degrees to radian measure?
To convert degree measure to radian measure, use the following steps:
To convert the degree measure into the radian measure, a simple formula can be applied. The formula is 1 rad= \[\frac{\pi}{180°}\]. With this, you can convert one unit of measure to another easily.
So, Multiply the given value with \[\frac{\pi}{180°}\].
\[ 24 \times \frac{\pi}{180^{0}} = \frac{2 \pi}{15} \] radians.
Therefore, the answer is \[\frac{2\pi}{15}\]radians.
5. What is a Degree?
It is used for the measurement of angles and is denoted by a little circle at the top of the number. ( 26°) Practical geometry involves the use of degrees to measure the angles. Since a circle rotates at 360°, one rotation gets divided into 360 parts. Each part is referred to as a degree. Therefore, angles can be measured at different degrees such as 30°, 45°, 60°, etc. The most convenient way to measure the angles in degrees is by a protractor. There are two sets of numbers on a protractor, one moves in a clockwise direction whereas the other one moves in the anti-clockwise direction. The outside curve is marked 0° to 180° from left to right. The inside curve is marked 180° to 0° from right to left. Readings on both the curves coincide to form 180°. The degree also has its subparts that are minutes and seconds.
6. Can the measure in degrees be negative?
Yes, it is possible. The degree measure can be negative. The negative sign with the value indicates that the angle is being measured in an opposite direction. The size of the angle whether with the positive or negative signs will remain the same. The sign depends on the direction of the rotation of the angle. The counterclockwise rotation results in a positive angle whereas the clockwise rotation results in a negative angle.
7. From where can we get the free study material and practice questions on Degrees and Radians?
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