Degrees to Radians Definitions and Examples

The term “radian” often comes up in mathematics and physics, but its meaning isn’t always immediately clear. In short, a radian is a unit of measurement for angles. One radian is defined as the angle between two line segments that intersect at the center of a circle and have lengths equal to the radius of that circle. While that definition may be helpful to some, it’s often easier to understand the concept with examples. In this blog post, we will provide several examples of degrees to radians conversions. We will also show how radians can be used in real-world scenarios, such as calculating the length of an arc on a circle.

Degrees to Radians

There are a few different ways to convert degrees to radians, but the most common is to simply multiply the number of degrees by pi/180. For example, 90 degrees would be equal to 90 x pi/180, or approximately 1.57 radians.

To convert degrees to radians, you can also use the following equation: radians = degrees x pi/180.

For example, if you wanted to find out what 45 degrees was in radians, you would plug 45 into the equation like so:

radians = 45 x pi/180

Which would give you an answer of approximately 0.79 radians.

What is Degrees to Radians Conversion?

Radians and degrees are two ways of measuring angles. A full circle is 360 degrees, which is also 2? radians. Because there are 2? radians in a circle, 1 radian is 180/? degrees. The degree to radian conversion formula is simple: multiply the number of degrees by ?/180. For example, 90° × ?/180 = ?/2 = 1.57 radians.

What are Degrees and Radians?

Radians and degrees are two ways of measuring angles. Both units are used in geometry and trigonometry, and they are often interchanged depending on the context or the needs of the problem.

Degrees are the most common unit for measuring angles, and they are denoted by the symbol °. There are 360° in a full circle, 180° in a half circle, 90° in a quarter circle, and so on. Degrees can be further divided into smaller units, like minutes and seconds. For example, 30° 30′ 30″ means 30 degrees + 30 minutes + 30 seconds.

Radians are denoted by the symbol rad, and they are defined as the ratio of the length of an arc to its radius. There are 2? radians in a full circle, ? radians in a half circle, ?/2 radians in a quarter circle, and so on. Because radians are based on ratios rather than fixed units like degrees, they can be converted to other units like gradians or turns.

Degrees to Radians Formula

The most common way to convert degrees to radians is by using the following formula:

radians = degrees x pi / 180

For example, if you want to convert 180 degrees to radians, you would plug those values into the formula like this:

radians = 180 x pi / 180

Which would give you a result of 3.14159, or pi. You can also use this calculator to convert degrees to radians.

There are a few different ways to remember the formula for converting degrees to radians. One way is to think about it in terms of a circle. There are 2pi radians in a full circle, so if you divide that by 360 degrees (the number of degrees in a circle), you get the following:

2pi radians / 360 degrees = pi / 180 degrees

You can also just memorize the formula, or make a note of it somewhere so that you can look it up when you need it.

Derivation of Degrees to Radians Formula

To derive the formula for converting degrees to radians, we start with the definition of a radian:

A radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle.

We can then use some basic trigonometry to derive the formula:

degree * pi/180 = radians

where pi is approximately 3.14159.

How to Convert Degrees to Radians?

Radians and degrees are units of angular measure. A degree is 1/360th of a circle, while a radian is the angle formed when the length of the radius of a circle is equal to the circumference of the circle. One degree is equivalent to 0.01745329252 radians.

To convert from degrees to radians, multiply by 0.01745329252. For example, 90° × 0.01745329252 = 1.57079632679 (radians).

To convert from radians to degrees, divide by 0.01745329252 or multiply by 57.2957795131 (the number of degrees in a radian). For example, 1.57079632679 (radians) ÷ 0.01745329252 = 90°

Degrees to Radians Conversion Table

A degree, usually denoted by ° (the degree symbol), is a unit of measurement of angles, one sixty-fourth of a turn. A radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. The relationship between degrees and radians is therefore:
$$1^{\circ} = \frac{\pi}{180}~\text{radians}$$
and
$$1~\text{radian} = \frac{180}{\pi}~\text{degrees}.$$
Here are some examples of common angles in both degrees and radians:

| Angle (degrees) | Angle (radians) |
|—————–|—————–|
| $90^{\circ}$ | $\frac{\pi}{2}$ |
| $180^{\circ}$ | $\pi$ |
| $360^{\circ}$ | $2\pi$ |

Degrees to Radians Chart

A degree is a unit of measurement that represents 1/360th of a circle. There are 360 degrees in a circle, and each degree is equal to radians. Radians are the SI unit for measuring angles, and they are defined as the ratio of the length of the arc to the radius of the circle.

There are 2? radians in a circle, and each radian is equal to 180/? degrees. Below is a chart that shows the conversion from degrees to radians:

Degrees Radians
0° 0 rad
1° 0.0174533 rad
2° 0.0349066 rad
3° 0.0523599 rad
4° 0.0698132 rad
5° 0.0872665 rad
6° 0.1047198rad
7° 0.122173rad
8° 0.1396263rad
9° 0.1570895rad
10° 0.174553rad
11° 0.1920164rad
12° 0.20947969

How to Convert Between the Two Units

When it comes to angles, there are two common units of measurement: degrees and radians. There are 360 degrees in a complete circle, while there are 2? radians in a complete circle.

To convert from degrees to radians, divide the number of degrees by 360 and multiply by 2?. For example, 45° would be (45/360) * 2? = ?/4 radians.

To convert from radians to degrees, divide the number of radians by 2? and multiply by 360. For example, ?/4 would be (?/4) / (2?) * 360 = 45°.

Examples of Degrees to Radians Conversions

There are 360 degrees in a circle. This means that there are 2? radians in a circle. Thus, one degree is equal to ?/180 radians.

To convert from degrees to radians, multiply the number of degrees by ?/180. For example, 45°×?/180=?/4radians.

Here are some more examples:
90°×?/180=?/2radians
120°×?/180=2?/3radians
240°×?/180=4?/3radians
360°×?/180=2?radians

When to Use Radians vs. Degrees

Radians and degrees are two units of measurement for angles. In general, radians are used when working with calculus or other mathematical concepts that require a more precise measure of an angle, while degrees are used in everyday situations.

There are 360 degrees in a circle, so one degree is 1/360th of the circumference. Similarly, there are 2? radians in a circle, so one radian is 1/2?th of the circumference. This means that one degree is equal to ?/180 radians, or about 0.0174533radians.

When working with calculus or other mathematical concepts, it is often easier to use radians rather than degrees because radians are a more precise measure of an angle. For example, consider the function y=sin(x). If we input x=1radian into this function, we will get y=0.8414709848… However, if we input x=57.29577951 degrees into this function (which is equivalent to 1radian), we will only get y=0.0174524064… Therefore, it is more accurate to use radians when working with this particular function.

Of course, there are many situations in which degrees are more appropriate than radians. In general, everyday situations call for the use of degrees rather than radians. For example, when describing the size of an angle made by two lines on a piece of paper,

Degrees to Radians Calculator

There are 360 degrees in a circle. To calculate the radian measurement of a degree, divide the number of degrees by 360. For example, 45° would be 45 divided by 360, or 0.125 radians.

You can use our online degrees to radians calculator to quickly and easily find the radian measurement for any degree value:

Just enter the degree measure into the field and click “Calculate”. The resulting radian value will be displayed in both decimal form and scientific notation.

So, what is a radian exactly? A radian is simply a unit of measurement for angles. One radian is equivalent to 57.2958 degrees (180/?). Radians are used in many mathematical calculations – especially in Trigonometry – so it’s important to know how to convert between them and degrees.

Now that you know how to use our calculator and what a radian is, let’s look at some examples of converting degrees to radians (and vice versa).

Conclusion

I hope this article has helped you understand the concept of radians and how to convert between degrees and radians. Radians are a convenient way to measure angles, especially in calculus, so it’s worth understanding how to work with them. If you need help converting between degrees and radians, remember that there are 2° in ? radians, or ? radians in 180°; also, 1 degree is equal to 0.017453292519943295radians. With a little practice, converting between these two units will become second nature. Thanks for reading!