# Degrees - Definition, Symbol, Conversion, and Examples

## What Is A Degree?

A **degree** is a unit of measurement for the size of an angle. One degree, written 1°, is defined as **1/360 of one full rotation** around a point. So a complete turn measures 360°, half a turn measures 180°, and a quarter turn measures 90°.

The small raised circle, °, is the **degree symbol**. It is written immediately after the number with no space: 45°, 90°, 360°. The symbol matters — "45°" means an angle, while "45" alone is just a count.

Degrees measure _how open_ an angle is, not how long its arms are. An angle of 30° stays 30° whether its arms are two centimetres long or two kilometres long. The arm length is irrelevant; only the amount of turn between the arms counts.

### The degree symbol and its subdivisions

For measurements finer than a whole degree, each degree splits further:

- **1 degree = 60 minutes of arc**, written 60′.
- **1 minute = 60 seconds of arc**, written 60″.

So a precise bearing might read 23° 30′ 15″. These subdivisions matter most in navigation, astronomy, and surveying, where a fraction of a degree can mean kilometres on the ground.

## How Do Degrees Relate To Radians?

Degrees are not the only way to measure angles — **radians** are the other main unit, and the two are linked by one anchor fact below:

180°=π radians

From this single equivalence, every conversion follows. To go from degrees to radians, multiply by \(\frac{\pi}{180}\). To go from radians to degrees, multiply by \(\frac{180}{\pi}\).

- A **radian** is the angle you get when the arc length along a circle equals the circle's radius. One full turn contains 2π radians, which is why 360°=2π radians.

- Degrees are intuitive for everyday measurement; radians are the natural unit for higher mathematics, because they make the formulas of trigonometry and calculus come out clean.

## Examples of Degrees

### Example 1

**How many degrees are there in three straight angles laid in a row?**

A straight angle is 180°. Three of them:

3×180°=540°

**Final answer:** 540°.

### Example 2

**Convert 60° to radians.**

Multiply by \(\frac{\pi}{180}\):

60°×\(\frac{\pi}{180}\) = \(\frac{60\pi}{180}\) = \(\frac{\pi}{3}\)

**Final answer:** \(\frac{\pi}{3}\) radians.

### Example 3

**A student converts 90° to radians and writes "90° = 90π radians." Where did it go wrong?**

_Wrong attempt._ The student remembers radians involve π and simply tacks π onto the number: 90°=90π.

_Why it breaks._ That answer is enormous — about 283 radians, more than 45 full turns — for an angle that is only a quarter turn. The error is skipping the conversion factor entirely.

_Correct._ Multiply by \(\frac{\pi}{180}\):

90°×\(\frac{\pi}{180}\) = \(\frac{90\pi}{180}\) = \(\frac{\pi}{2}\) radians

**Final answer:** \(\frac{\pi}{2}\) radians.

### Example 4

**Two angles measure 47° 30′ and 12° 45′. Add them.**

Add minutes and degrees separately:

Minutes: 30′ + 45′ = 75′ = 1° 15′

Degrees: 47° + 12° + 1° = 60°

**Final answer:** 60° 15′.

### Example 5

**A clock's minute hand moves from 12 to 3. Through how many degrees does it turn?**

The clock face is a full turn, 360°, split into 12 equal hour marks. From 12 to 3 is three marks:

360°/12 × 3 = 90°

**Final answer:** 90°.

### Example 6

**Convert \(\frac{3\pi}{4}\) radians to degrees.**

Multiply by \(\frac{180}{\pi}\):

\(\frac{3\pi}{4} \times \frac{180}{\pi} = \frac{3 × 180}{4} = 135°\)

**Final answer:** 135°.

## Why We Measure Angles In Degrees At All

The degree is so familiar that its usefulness is easy to overlook. Naming what it does shows why it has survived for thousands of years.

- **It makes angles comparable.** Without a shared unit, "a sharp angle" and "a wide angle" are just opinions. Degrees turn them into 30° and 150° — values you can add, subtract, and reason about precisely.

- **360 was chosen for its divisibility.** Because 360 splits evenly into many numbers, fractions of a circle land on whole numbers.

- **It runs the physical world.** Compass bearings, the tilt of a roof, and the steering lock of a car — all are quoted in degrees.

- **It is the bridge to higher math.** Degrees connect to radians and into trigonometry and calculus.

## Where Students Trip Up With Degrees

### Mistake 1: Converting to radians by just attaching π

- **Don't do this:** Writing "60° = 60π".
- **The correct way:** Multiply by \(\frac{\pi}{180}\).

### Mistake 2: Dropping or misplacing the degree symbol

- **Don't do this:** Omitting the ° symbol.
- **The correct way:** Attach ° directly to every angle value: 45°, 90°, etc.

### Mistake 3: Confusing arm length with angle size

- **Don't do this:** Judging which angle is larger by how long its arms are drawn.
- **The correct way:** The degree measure depends only on how open the angle is.

## Conclusion

- A **degree (°)** is a unit of angle measure equal to 1/360 of a full turn.
- The degree measure of an angle depends only on how open it is.
- Degrees convert to radians by multiplying by \(\frac{\pi}{180}\).

## Practice These To Solidify Your Understanding

Work through these:

1. Convert 45° to radians. 
2. Through how many degrees does a clock's minute hand turn from 12 to 6? 
3. Convert \(\frac{2\pi}{3}\) radians to degrees.