When we work with angles, it's important to be able to express to others the *measure* of an angle. How sharp is a corner? Do two lines form a *narrow* or *wide* angle, and precisely *how* narrow or wide?

We mostly use three systems of angle measurement in mathematics and every-day life. They are:

The last is related to the first, but we'll cover it in a different section because it's mainly used for navigation, and doesn't arise too much in math and science.

At the end of this section, we'll touch on the idea of a solid (3-D) angle, measured in **steradians**.

Always remember that expressing the measure of an angle in different units doesn't change the measure of the angle (see unit conversion). We can say, for example, that Mt. Everest is 29,028 ft. high or that it is 8848 meters high. Both are true, and no adjustment of the actual altitude of Mt. Everest was necessary. It stayed the same.

The **degree** (the unit is given the symbol **˚** ) is the most commonly used measurement of angle. It's the one we learn as young kids and we all more-or-less have a good feeling for angles measured in degrees.

For example, we're familiar with **right angles**, which measure 90 degrees (90˚) as the common angle in most construction projects, including the angles between the walls or walls and ceilings of most buildings. The inside angles of a square or rectangle are right angles. When two lines are said to be "square," we mean they form a right angle.

In mathematics, we describe angles on the **unit circle** (below), a circle that begins in the first quadrant of an x-y graph, and sweeps counterclockwise around to form a circle.

The unit circle, so named because the "dial" that sweeps around to form any angle is one unit long, shows that moving that dial (or vector) around the center point back to where it began covers 360˚ of arc or angle. That means that half of a circle is 180˚, a quarter-circle is 90˚, and so on.

A circle comprises 360 degrees of arc.

Another use of the unit circle is the **compass rose**, like what you would see on the face of a compass. (Older compasses had a star or flower-like design on their faces, thus "rose.")

On a compass rose, we begin our angles at north, or 0˚ on the compass. East is 90˚, south is 180˚, and west is 270˚ from north. North, south, east and west are called the **cardinal** points of a compass. The points between those (45˚ away) are called the **ordinal** points (or sometimes inter cardinal points), northeast (NE), southeast (SE), northwest (NW) and southwest (SW).

Airplanes and ships make use of the compass to stay on a particular course. The runways of an airport, in fact, are labeled by which direction a plane travels in order to land squarely on them. The runways below, labeled 27 L and 27 R are oriented at about 270˚ to north. That is, a plane landing on either would land from the west toward the east. The L and R simply differentiate the two parallel runways, as often exist at large airports.

Runways are labeled by finding the approach bearing (angle), rounding it to the nearest 10˚ and dropping the trailing zero. For example, a runway oriented for landing from an angle of 82˚ from north would be rounded to 80˚ and labeled runway 8 — just a quirk of the aviation world, but now you know what those labels mean!

Why did someone long ago decide that a circle should be divided into **360** chunks (degrees) rather than, say, **100**?

It turns out that numbers like 360˚ and the 60-minute hour likely arose because of the convenience of using fractions of those numbers. While 100 can be divided into 50^{th}s, 10^{th}s, and so on, 60 can be divided in half, 1/3, 1/4, 1/6, 1/10, 1/12, and a few more convenient fractions. It made a lot of sense before the digital age.

Number of degrees | Circle fraction |
---|---|

360 | 1 |

180 | ½ |

120 | ⅓ |

90 | ¼ |

60 | ⅙ |

45 | ⅛ |

30 | 1/12 |

15 | 1/24 |

10 | 1/36 |

5 | 1/72 |

**Radians** is the *natural* unit of angle measurement, and a much more useful unit in mathematics. To see how it arises, we begin with the definition of **pi** (**π**), the ratio of the diameter to the circumference of *any *circle:

That ratio is constant for circles of any size, and yields the familiar irrational number, **π = 3.14159...** Because the diameter of a circle is **d = 2r**, where **r** is the **radius**, we also have

$$\pi = \frac{c}{2r}$$

We can solve for the circumference like this:

$$c = 2 \pi r$$

Now if we take a **unit circle** (**r** = 1), we have $c = 2 \pi$, the number of radians in a circle. A radian is defined as the length of a circular arc divided by the radius of that arc. In the case of our unit circle, that's $2 \pi$ radians, so just as there are 360˚ in the arc of a complete circle, so there are $2\pi$ *radians* of arc.

Now $2\pi$ radians is something like 6.283 radians, but in mathematics, it's very common not to multiply by $\pi$ explicitly, and just leave 6.283... as $2\pi$. We also abbreviate radians to "**rad**."

Here are some common angles in degrees and radians:

Degrees | Radians (π) | ≈ Radians |
---|---|---|

360˚ | 2π rad | 6.283 rad |

180˚ | π rad | 3.142 rad |

90˚ | π/2 rad | 1.571 rad |

60˚ | π/3 rad | 1.05 rad |

45˚ | π/4 rad | 0.785 rad |

30˚ | π/6 rad | 0.524 rad |

To convert between degrees and radians, we need only know one fact that relates the two. A convenient one is that $\pi \, \text{rad} = 180^{\circ}$. From that fact, all conversions are easy. Here are a couple of examples

Convert 230˚ to radians, and express the result as a multiple of π.

**Solution**

$$ \begin{align} 230˚ \left( \frac{\pi \, rad}{180˚} \right) &= 1.28 \pi \, \text{rad} \\[5pt] &= 4.01 \, \text{rad} \end{align}$$

Either way of expressing the result is fine, but many times in math and science, π's end up canceling, so it's often useful not to multiply by π = 3.14159 ... until it's necessary.

Convert 0.9 rad to degrees and express the result to the nearest tenth of a degree.

**Solution**

$$ \require{cancel} 0.9 \, \cancel{rad} \left( \frac{180˚}{\pi \, \cancel{rad}} \right) = 51.6˚$$

When converting from radians to degrees, go ahead and divide by π = 3.14159... rad to get a number of degrees.

Convert the following angle measures to degrees.

1. | $2.92 \text{ rad}$ | |

2. | $6 \pi \text{ rad}$ |

3. | $\pi /4.2 \text{ rad}$ | |

4. | $63 \pi / 11 \text{ rad}$ |

Convert the following angle measures to radians.

5. | 310˚ | |

6. | 118˚ |

7. | 720˚ | |

8. | 12˚ |

While we're talking about angles, we should probably touch on the concepts of minutes and seconds as a fraction of a degree. The **degrees-minutes-seconds** system is sometimes referred to as DMS. It is used a lot in navigation, namely in quoting latitudes and longitudes to find a position on the globe.

**Minutes** and **seconds** (in the context of degrees) are just smaller divisions of the degree.

- 1 degree = 60 minutes
- 1 minute = 60 seconds
- ... so 1˚ = 3600 s, just as 1 hour = 3600 s

Sometimes we refer to the minute and second as "minutes of arc" and "seconds of arc," respectively.

In global navigation, any coordinate on the globe is referred to by a combination of two angles, one north or south from the equator (0˚), called the latitude, and one east or west from the prime meridian, a line passing through (for historical reasons) the town of Greenwich, England.

For example, the coordinates of the center of Portland, OR are 45˚ 30' 44.028"N and 122˚ 39' 31.39" W, where **'** stands for minutes, **"** stands for seconds, and N and W stand for north or south of the equator (for longitude) and east or west of the prime meridian (the agreed-upon zero-degree east-west line), respectively.

The **nautical mile** (about 1.15 miles) was originally defined as the distance of 1 minute of latitude arc on Earth. In that way, the nautical mile is more directly tied to the size of something relevant (Earth) than the mile, which was somewhat arbitrarily defined originally

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