Angle Calculator
Angle Calculator
Calculate angles, convert between degrees, radians, gradians, find complementary/supplementary angles, and solve trigonometry problems.
Angle Calculation Result
0.7854 rad
Calculation Method:
Step-by-Step Calculation:
Angle Analysis:
Angle Visualization:
Angle measurement and conversion between different units.
What is an Angle?
An angle is a geometric figure formed by two rays (or line segments) sharing a common endpoint called the vertex. Angles measure the amount of rotation between the two rays and are fundamental in geometry, trigonometry, physics, and engineering.
Angle Measurement Systems
Degrees (°)
Most common system
Everyday use
Radians (rad)
Mathematical standard
Calculus & physics
Gradians (gon)
Surveying & engineering
Decimal system
Revolutions (rev)
Rotational motion
Full circles
Angle Conversion Formulas
1. Basic Conversion Formulas
Convert between different angle units:
• Degrees to Radians: rad = deg × (π/180)
• Radians to Degrees: deg = rad × (180/π)
• Degrees to Gradians: gon = deg × (400/360)
• Gradians to Degrees: deg = gon × (360/400)
2. Special Angle Relationships
Important angle properties:
• Complementary angles: α + β = 90°
• Supplementary angles: α + β = 180°
• Coterminal angles: differ by multiples of 360°
• Reference angle: acute angle to x-axis
3. Common Angle Values
Frequently used angles:
• 30° = π/6 rad ≈ 0.5236 rad
• 45° = π/4 rad ≈ 0.7854 rad
• 60° = π/3 rad ≈ 1.0472 rad
• 90° = π/2 rad ≈ 1.5708 rad
Angle Types by Measure
| Angle Type | Range | Examples | Properties |
|---|---|---|---|
| Acute Angle | 0° < θ < 90° | 30°, 45°, 60° | Less than right angle |
| Right Angle | θ = 90° | 90° = π/2 rad | Forms L shape, perpendicular |
| Obtuse Angle | 90° < θ < 180° | 120°, 135°, 150° | Greater than right angle |
| Straight Angle | θ = 180° | 180° = π rad | Forms straight line |
| Reflex Angle | 180° < θ < 360° | 200°, 270°, 320° | Greater than straight angle |
| Full Rotation | θ = 360° | 360° = 2π rad | Complete circle |
Common Angles and Their Values
| Degrees | Radians | Gradians | Sine | Cosine | Tangent |
|---|---|---|---|---|---|
| 0° | 0 | 0 gon | 0 | 1 | 0 |
| 30° | π/6 | 33.33 gon | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | 50 gon | √2/2 | √2/2 | 1 |
| 60° | π/3 | 66.67 gon | √3/2 | 1/2 | √3 |
| 90° | π/2 | 100 gon | 1 | 0 | ∞ |
| 180° | π | 200 gon | 0 | -1 | 0 |
| 270° | 3π/2 | 300 gon | -1 | 0 | ∞ |
| 360° | 2π | 400 gon | 0 | 1 | 0 |
Real-World Applications
Engineering & Architecture
- Structural design: Calculating roof pitches and support angles
- Mechanical engineering: Gear tooth angles and cam profiles
- Civil engineering: Road gradients and bridge inclinations
- Surveying: Measuring land boundaries and topographic angles
Physics & Astronomy
- Projectile motion: Launch angles for maximum range
- Optics: Reflection and refraction angles (Snell's Law)
- Astronomy: Celestial coordinates and planetary orbits
- Mechanics: Force vectors and torque calculations
Computer Graphics & Gaming
- 3D modeling: Vertex angles and surface normals
- Game development: Character rotation and camera angles
- Animation: Joint angles for character rigging
- Computer vision: Object orientation and perspective
Everyday Life
- Navigation: Compass bearings and GPS coordinates
- Sports: Golf club angles, basketball shot angles
- Photography: Camera angles and field of view
- Carpentry: Cutting angles for joints and frames
Trigonometric Functions
| Function | Definition | Range | Period | Example (45°) |
|---|---|---|---|---|
| Sine (sin) | opposite/hypotenuse | [-1, 1] | 2π (360°) | sin(45°) = √2/2 ≈ 0.7071 |
| Cosine (cos) | adjacent/hypotenuse | [-1, 1] | 2π (360°) | cos(45°) = √2/2 ≈ 0.7071 |
| Tangent (tan) | opposite/adjacent | (-∞, ∞) | π (180°) | tan(45°) = 1 |
| Cosecant (csc) | 1/sin | (-∞, -1] ∪ [1, ∞) | 2π (360°) | csc(45°) = √2 ≈ 1.4142 |
| Secant (sec) | 1/cos | (-∞, -1] ∪ [1, ∞) | 2π (360°) | sec(45°) = √2 ≈ 1.4142 |
| Cotangent (cot) | 1/tan | (-∞, ∞) | π (180°) | cot(45°) = 1 |
Step-by-Step Angle Calculation Process
Example 1: Convert 45° to Radians
- Use conversion formula: radians = degrees × (π/180)
- Substitute: radians = 45 × (π/180)
- Simplify: 45/180 = 1/4
- Calculate: radians = (1/4)π ≈ 0.7854
- Result: 45° = π/4 rad ≈ 0.7854 rad
Example 2: Find Complementary Angle of 30°
- Complementary angles sum to 90°
- Formula: complementary = 90° - given angle
- Calculate: 90° - 30° = 60°
- Check: 30° + 60° = 90° ✓
- Result: Complementary angle is 60°
Example 3: Calculate sin(60°)
- Recognize 60° as special angle
- From trigonometric table: sin(60°) = √3/2
- Calculate approximate value: √3/2 ≈ 1.732/2 ≈ 0.8660
- Interpretation: In 30-60-90 triangle, opposite side is √3 times shorter side
- Result: sin(60°) = √3/2 ≈ 0.8660
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Frequently Asked Questions (FAQs)
Q: What's the difference between degrees and radians?
A: Degrees divide a circle into 360 parts, while radians use the radius as a unit (1 radian = angle where arc length equals radius). Radians are mathematically more natural for calculus and physics, while degrees are more intuitive for everyday use.
Q: How do you find coterminal angles?
A: Coterminal angles differ by multiples of 360° (or 2π radians). To find coterminal angles, add or subtract 360° (or 2π) from the given angle. Example: 45° and 405° are coterminal (45° + 360° = 405°).
Q: What is a reference angle?
A: A reference angle is the acute angle (0° to 90°) formed between the terminal side of an angle and the x-axis. It's always positive and helps find trigonometric values for angles in any quadrant.
Q: Why are 30°, 45°, and 60° special angles?
A: These angles have exact trigonometric values that can be expressed using square roots and rational numbers, making them fundamental in trigonometry. They come from equilateral and right isosceles triangles.
Master angle calculations with Toolivaa's free Angle Calculator, and explore more mathematical tools in our Math Calculators collection.