please explain differences between turns radians degrees gradians
[Angle Measurment Units Comparison]
These units are all ways to measure angles, each suited to different applications and systems. Here's a breakdown of their differences:
- Definition: A "turn" is a complete rotation around a circle, representing one full cycle.
- Relation to Circle: One turn = 360 degrees.
- Conversion:
- 1 turn = (2\pi) radians.
- 1 turn = 400 gradians.
- 1 turn = 360 degrees.
- Use: Often used in computer graphics, animations, and intuitive descriptions of rotations.
- Definition: A radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
- Relation to Circle: (2\pi) radians = 1 full turn.
- Conversion:
- (1 \text{ radian} = \frac{180}{\pi} \approx 57.2958) degrees.
- (1 \text{ radian} = \frac{200}{\pi} \approx 63.662) gradians.
- (2\pi \text{ radians} = 1) turn.
- Use: Common in mathematics and physics, particularly in calculus and trigonometry, because they simplify formulas.
- Definition: A degree divides a circle into 360 equal parts, with one part representing 1 degree.
- Relation to Circle: 360 degrees = 1 full turn.
- Conversion:
- (1 \text{ degree} = \frac{\pi}{180} \approx 0.01745) radians.
- (1 \text{ degree} = \frac{10}{9} \approx 1.1111) gradians.
- (360 \text{ degrees} = 1) turn.
- Use: Widely used in everyday applications, navigation, engineering, and astronomy.
- Definition: A gradian divides a circle into 400 equal parts, with one part representing 1 gradian.
- Relation to Circle: 400 gradians = 1 full turn.
- Conversion:
- (1 \text{ gradian} = \frac{\pi}{200} \approx 0.015708) radians.
- (1 \text{ gradian} = 0.9) degrees.
- (400 \text{ gradians} = 1) turn.
- Use: Used primarily in surveying, engineering, and some European countries.
| Unit | Turns | Radians | Degrees | Gradians |
|---|---|---|---|---|
| 1 Turn | 1 | (2\pi) | 360 | 400 |
| 1 Radian | (1/2\pi) | 1 | 57.2958 | 63.662 |
| 1 Degree | (1/360) | (\pi/180) | 1 | 1.1111 |
| 1 Gradian | (1/400) | (\pi/200) | 0.9 | 1 |
Each unit is most effective in specific contexts, so the choice depends on the task or convention being followed.