# Radians

We can measure Angles in Radians. 1 Radian is |

Does 57.2958... degrees seem a strange value?

Maybe degrees are strange, as the **Radian** is a pure measure based on the **Radius** of the circle:

**Radian**: the angle made by taking the **radius**

and **wrapping it along the edge** of a circle:

So, a Radian "cuts out" a length of a circle's circumference equal to the radius. |

## Radians and Degrees

So:

- There are π radians in a half circle
- And also 180° in a half circle

So π radians = 180°

So 1 radian = 180°/π = 57.2958° (approximately)

Degrees | Radians (exact) |
Radians (approx) |
---|---|---|

30° | π/6 | 0.524 |

45° | π/4 | 0.785 |

60° | π/3 | 1.047 |

90° | π/2 | 1.571 |

180° | π | 3.142 |

270° | 3π/2 | 4.712 |

360° | 2π | 6.283 |

### Example: How Many Radians in a Full Circle?

Imagine you cut up pieces of string exactly the length from the **center of a circle to its edge** ...

... how many pieces would you need to go **around the edge** of the circle?

Answer: 2π (or about **6.283** pieces of string).

## Radians Preferred by Mathematicians

Because the radian is based on the pure idea of *"the radius being laid along the circumference*", it often gives simple and natural results when used in mathematics.

**For example**, look at the sine function for very small values:

x (radians) | 1 | 0.1 | 0.01 | 0.001 |
---|---|---|---|---|

sin(x) | 0.8414710 | 0.0998334 | 0.0099998 | 0.0009999998 |

(as long as "x" is in Radians!)

There will be other examples like that as you learn more about mathematics.

## Conclusion

So, degrees are easier to use in everyday work, but radians are much better for mathematics.