# Radians

*The angle made when the radius
is
wrapped round the circle:*

1 Radian is |

*Why "57.2958..." degrees?* We will see in a moment.

The **Radian** is a pure measure based on the **Radius** of the circle:

**Radian**: the angle made when we take the **radius**

and wrap it round the circle.

## Radians and Degrees

Let us see why 1 Radian is equal to 57.2958... degrees:

In a half circle there are π radians, which is also 180°

**1 radian**= 180°/π

To go from **radians to degrees**: multiply by 180, divide by π

To go from **degrees to radians**: multiply by π, divide by 180

Here is a table of equivalent values:

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 pieces of string exactly the length from the **center to the circumference of a circle** ...

... how many pieces do you need to go **once around** 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.

## Small Angles

For small angles the values of the sine and tangent functions get close to the value of the angle in radian:

x (radians) | sin(x) | tan(x) |
---|---|---|

1 | 0.8414710 | 1.55740772 |

0.1 | 0.0998334 | 0.1003347 |

0.01 | 0.0099998 | 0.0100003 |

Here we can see the tan function on a triangle for smaller and smaller angles:

At 0.01 radians both sin and tan are within 0.003% of the radian value.

### Example: A road rises 1 m for every 100 m along. What is it's angle in degrees (without using a calculator)?

1 in 100 is 0.01, and tan(0.01) is approximately 0.01 radians

We also know that 1 radian is about 57 degrees, so 0.01 radians is about 0.57 degrees

Also the cosine function gets close to 1 for small radian values.

## Conclusion

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