Convert angle measurements between degrees and radians using the Degrees to Radians converter. Easily switch between degrees and radians by entering the value in the appropriate field and clicking the "Convert" button.

A radian is a unit of measurement for angles, often used in mathematics and physics. It is defined as the angle subtended by an arc of a circle that is equal in length to the radius of the circle. In other words, if a circle has a radius of r, then an angle of 1 radian corresponds to an arc length of r along the circumference of the circle. The radian is a dimensionless unit, and it is widely used in trigonometry and calculus due to its natural relationship with the properties of circles and angles.

A degree is a unit of measurement for angles, denoted by the symbol "°". It is commonly used to measure and express the size or extent of an angle. One complete revolution or circle is divided into 360 degrees. Each degree is further divided into 60 minutes (denoted as ') and each minute is divided into 60 seconds (denoted as ").

The degree measurement system is based on dividing a circle into equal parts, with each part representing one degree. It is widely used in various fields such as mathematics, engineering, navigation, and geography to quantify and describe the angular relationships between objects or points.

For example, a right angle is defined as 90 degrees (90°), while a straight line measures 180 degrees (180°), and a full circle measures 360 degrees (360°). Degrees are used to specify angles in geometric shapes, trigonometric functions, and navigation systems.

To convert from degrees to radians, you can use the following formula:

Radians = (Degrees * π) / 180

In this formula, "Degrees" represents the angle measurement in degrees, and "π" (pi) is a mathematical constant approximately equal to 3.14159.

To convert degrees to radians, multiply the angle in degrees by π and then divide the result by 180.

For example, let's say you have an angle of 45 degrees. The conversion to radians would be:

Radians = (45 * π) / 180

Simplifying the equation, we get:

Radians = π/4

So, 45 degrees is equivalent to π/4 radians.

Note that when using calculators or programming languages, there are often built-in functions or constants to perform the conversion between degrees and radians. For instance, in many programming languages, you can use the "radians()" function to convert degrees to radians.

To convert from radians to degrees, you can use the following formula:

Degrees = (Radians * 180) / π

In this formula, "Radians" represents the angle measurement in radians, and "π" (pi) is a mathematical constant approximately equal to 3.14159.

To convert radians to degrees, multiply the angle in radians by 180 and then divide the result by π.

For example, let's say you have an angle of π/4 radians. The conversion to degrees would be:

Degrees = (π/4 * 180) / π

Simplifying the equation, we get:

Degrees = 45

So, π/4 radians is equivalent to 45 degrees.

Similarly to converting from degrees to radians, many calculators and programming languages provide built-in functions or constants to perform the conversion between radians and degrees. For instance, in many programming languages, you can use the "degrees()" function to convert radians to degrees.

Why are radians used as a unit of measurement for angles? Radians are a fundamental unit of angle measurement that offer certain advantages in mathematical calculations and trigonometry. Unlike degrees, which are based on dividing a circle into 360 equal parts, radians are based on dividing a circle into equal parts based on the length of its radius.

An angle of 1 radian is defined as the angle that subtends an arc on a circle equal in length to the radius of the circle. This relationship between the angle and the arc length simplifies many trigonometric calculations and allows for more precise and consistent results.

name of the angle | value in radians | value in degrees |

zero angle | 0 rad | 0° |

milliradian | 1 mrad | 0°3′26″15‴ |

π/6 rad | 30 ° | |

π/4 rad | 45° | |

π/3 rad | 60 ° | |

radian | 1 rad | 57°17′44″48‴ |

right angle | π/2 rad | 90° |

2π/3 rad | 120 ° | |

3π/4 rad | 135° | |

flat angle | π rad | 180° |

5π/4 rad | 225° | |

3π/2 rad | 270° | |

7π/4 rad | 315° | |

full angle | 2π rad | 360° |

By using radians, mathematical operations involving angles become more straightforward and compatible with other mathematical functions and formulas. Radians also provide a more intuitive representation of angles in many mathematical and scientific disciplines, making calculations and conversions more convenient and accurate.

When dealing with angles of small values, certain approximations can be made to simplify calculations.

For an angle with a value less than 0.17 radians (approximately 10 degrees), the error resulting from approximating the angle is less than 1%. This means that the calculated value using the approximation is within 1% of the exact value.

For an even smaller angle with a value less than 0.05 radians (approximately 3 degrees), the error is further reduced to less than 0.1%. This indicates that the approximation is even more accurate, with the calculated value being within 0.1% of the precise value.

These approximations are commonly used when working with small angles in various fields, such as mathematics, physics, and engineering. They provide a practical way to simplify calculations without introducing significant errors, especially when the precise measurement of small angles is not necessary for the intended analysis or application.

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