# Radians

In a unit circle, a radian is the angle subtended by an arc whose length is equal to the radius.

One radian is approximately **57.3°** in the degree system.

Radians are the unit of measurement for angles.

They are denoted by the symbol rad.

**Note**: There are other units for measuring angles. Unless otherwise specified, it is assumed that the measurement is in radians even if the rad symbol is omitted.

## Proof

Consider the same angle α in two circles with different radii, r and r'.

If I measured the size of the angles by the length of the arc, I would get two different lengths.

For the same angle, the second circle has a longer arc (k'>k) than the first circle.

According to the proportionality between angles and arcs:

$$ k : \alpha = 2 \pi r : 360° $$

$$ k' : \alpha = 2 \pi r' : 360° $$

This means:

$$ \frac{k}{\alpha} = \frac{2 \pi r}{360°} $$

$$ \frac{k'}{\alpha} = \frac{2 \pi r'}{360°} $$

Highlighting the arc lengths:

$$ k = \frac{2 \pi r \cdot \alpha}{360°} $$

$$ k' = \frac{2 \pi r' \cdot \alpha}{360°} $$

Simplifying 2 and 360:

$$ k = \frac{\pi r \alpha}{180°} $$

$$ k' = \frac{\pi r' \alpha}{180°} $$

Dividing both sides:

$$ \frac{k}{k'} = \frac{\frac{\pi r \alpha}{180°}}{\frac{\pi r' \alpha}{180°}} $$

$$ \frac{k}{k'} = \frac{\pi r \alpha}{180°} \cdot \frac{180°}{\pi r' \alpha} $$

$$ \frac{k}{k'} = \frac{r}{r'} $$

By moving the quantities of the first circle to the left and those of the second circle to the right:

$$ \frac{k}{r} = \frac{k'}{r'} $$

We notice that the ratio between the length of the arc and the radius is the same in both circles.

$$ k : r = k' : r' $$

This means the ratio between the arc and the radius (k/r) does not change with the size of the circle.

Therefore, the k/r ratio is a reliable unit of measurement for the size of an angle because it is universal and allows for comparison of angles from circles with different radii.

$$ \alpha = \frac{k}{r} $$

The k/r ratio is called the **radian** (rad) and is equal to 1 when the length of the arc k is equal to the length of the radius r.

## The Measurement of Key Angles in Radians and Degrees

This table summarizes the key angle measurements in radians and degrees:

Degrees | Radians |
---|---|

0° | 0 |

15° | π/12 |

30° | π/6 |

45° | π/4 |

60° | π/3 |

90° | π/2 |

120° | 2/3 π |

135° | 3/4 π |

150° | 5/6 π |

180° | π |

270° | 3/2 π |

360° | 2 π |

**How to Remember Them All? **Initially, this can be challenging. However, if you remember that $$ 15° = \frac{\pi}{12} \ rad $$ you can reconstruct all the others: $$ 30° = 2 \cdot 15° = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6} \ rad $$ $$ 45° = 3 \cdot 15° = 3 \cdot \frac{\pi}{12} = \frac{\pi}{4} \ rad $$ $$ 60° = 4 \cdot 15° = 4 \cdot \frac{\pi}{12} = \frac{\pi}{3} \ rad $$ $$ 90° = 6 \cdot 15° = 6 \cdot \frac{\pi}{12} = \frac{\pi}{2} \ rad $$ Alternatively, for more precise calculations, remember that 1° equals $$ 1° = \frac{2\pi}{360} = \frac{\pi}{180} \ rad $$ And so on.

### Why is a Full Circle 2π Radians?

Consider the entire circumference, a full circle, which is a 360° angle.

From geometry, we know that the length of a circumference is equal to the radius r multiplied by 2 pi (π).

$$ k = 2 \pi r $$

Knowing the length of the circumference (k) and the radius (r), we can calculate the angle in radians (rad) as:

$$ \alpha = \frac{k}{r} \ rad $$

Substituting k with 2πr, we get:

$$ \alpha = \frac{2 \pi r}{r} \ rad $$

Simplifying, we find the measure of the circumference in radians:

$$ \alpha = 2 \pi \ rad $$

Knowing that pi (π) equals 3.14:

$$ \alpha = 2 \cdot 3.14 \ rad = 6.28 \ rad $$

In conclusion, a full circle (360°) measures 2π radians, or **6.28 radians**.

**Note**: If a full circle (360°) equals 2π radians, a straight angle (180°) is π radians, or 3.14 radians, since it is exactly half. Therefore, a right angle (90°) is π/2 radians. And so on.

And so forth.