Angles — Complete Guide
An angle is formed when two rays share a common endpoint (vertex). Angles are measured in degrees (°) or radians (rad). One full rotation = 360° = 2π radians.
Types of Angles
| Type | Range | Example |
|---|---|---|
| Zero angle | 0° | No rotation |
| Acute | 0° < θ < 90° | 45°, 30°, 60° |
| Right | θ = 90° | Corner of a square |
| Obtuse | 90° < θ < 180° | 120°, 150° |
| Straight | θ = 180° | Flat line |
| Reflex | 180° < θ < 360° | 270°, 300° |
| Full rotation | θ = 360° | Complete circle |
Degrees vs Radians Conversion
Radians = Degrees × π / 180
Degrees = Radians × 180 / π
Key: 180° = π radians | 90° = π/2 | 360° = 2π
Degrees = Radians × 180 / π
Key: 180° = π radians | 90° = π/2 | 360° = 2π
| Degrees | Radians | Exact Radians |
|---|---|---|
| 30° | 0.5236 | π/6 |
| 45° | 0.7854 | π/4 |
| 60° | 1.0472 | π/3 |
| 90° | 1.5708 | π/2 |
| 120° | 2.0944 | 2π/3 |
| 180° | 3.1416 | π |
| 270° | 4.7124 | 3π/2 |
| 360° | 6.2832 | 2π |
Angle Pairs
Complementary angles: A + B = 90°
Supplementary angles: A + B = 180°
Triangle angles: A + B + C = 180°
Quadrilateral: sum = 360°
Regular polygon (n sides): each interior angle = (n−2)×180°/n
Supplementary angles: A + B = 180°
Triangle angles: A + B + C = 180°
Quadrilateral: sum = 360°
Regular polygon (n sides): each interior angle = (n−2)×180°/n
Interior Angles of Polygons
| Shape | Sides | Sum of Interior Angles | Each Angle (regular) |
|---|---|---|---|
| Triangle | 3 | 180° | 60° |
| Square/Rect | 4 | 360° | 90° |
| Pentagon | 5 | 540° | 108° |
| Hexagon | 6 | 720° | 120° |
| Octagon | 8 | 1080° | 135° |
| Decagon | 10 | 1440° | 144° |
Real-World Applications
- Construction: Roof pitch, staircase angles, door hinge angles
- Navigation: Compass bearings, flight paths, ship headings
- Astronomy: Angular separation of stars, telescope alignment
- Sports: Launch angles in projectile sports (javelin, shot put, golf)