Angle Calculator | Degrees Radians Gradians Converter UK 2025
Free Angle | Degrees Radians Gradians Converter UK 2025 calculator for the UK. Get instant, accurate results with our easy-to-use online tool. Updated for...
Last updated: February 2026
Angle Calculator
Convert between degrees, radians, gradians and turns. Calculate trigonometric functions and related angles instantly.
Angle Conversions
Trigonometric Functions
Related Angles
How to Use the Angle Calculator
Our angle calculator makes it easy to convert between different angle units and calculate trigonometric values:
- Enter your angle value in the input field, or click a quick angle button
- Select the unit of your input (degrees, radians, gradians, or turns)
- View instant results including all conversions, trig functions, and related angles
Quick Tip
Click the quick angle buttons (30°, 45°, 60°, etc.) to instantly see common angle values used in geometry and trigonometry.
Understanding Angle Units
Degrees (°)
The most common angle unit. A full circle contains 360 degrees. Degrees are used in everyday applications, navigation, and most educational contexts in the UK.
Radians (rad)
The standard unit in mathematics and physics. A full circle equals 2π radians (approximately 6.283). Radians are essential for calculus, engineering, and scientific calculations.
Gradians (gon)
Also known as gon or grade. A full circle contains 400 gradians, making a right angle exactly 100 gradians. Used primarily in surveying, civil engineering, and land measurement.
Turns
The simplest angle unit. One turn equals one complete rotation (360° or 2π radians). Useful for describing rotations and cycles.
Conversion Formulas
Common Angle Conversions
Reference table for frequently used angle values in mathematics and engineering:
| Degrees | Radians | Gradians | Turns | Type |
|---|---|---|---|---|
| 0° | 0 | 0 gon | 0 | Zero angle |
| 30° | π/6 ≈ 0.524 | 33.33 gon | 1/12 | Acute |
| 45° | π/4 ≈ 0.785 | 50 gon | 1/8 | Acute |
| 60° | π/3 ≈ 1.047 | 66.67 gon | 1/6 | Acute |
| 90° | π/2 ≈ 1.571 | 100 gon | 1/4 | Right angle |
| 120° | 2π/3 ≈ 2.094 | 133.33 gon | 1/3 | Obtuse |
| 180° | π ≈ 3.142 | 200 gon | 1/2 | Straight angle |
| 270° | 3π/2 ≈ 4.712 | 300 gon | 3/4 | Reflex |
| 360° | 2π ≈ 6.283 | 400 gon | 1 | Full rotation |
Trigonometric Values for Special Angles
Exact values for trigonometric functions at commonly used angles:
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 = 0.5 | √3/2 ≈ 0.866 | 1/√3 ≈ 0.577 |
| 45° | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 | 1 |
| 60° | √3/2 ≈ 0.866 | 1/2 = 0.5 | √3 ≈ 1.732 |
| 90° | 1 | 0 | undefined (∞) |
| 180° | 0 | -1 | 0 |
| 270° | -1 | 0 | undefined (∞) |
Important Note
Tangent (tan) and cotangent (cot) are undefined at angles where their denominators equal zero. Tan is undefined at 90°, 270°, etc., while cot is undefined at 0°, 180°, etc.
Types of Angles
Understanding the classification of angles is essential for geometry and practical applications:
| Angle Type | Range | Description | Example Use |
|---|---|---|---|
| Acute | 0° < θ < 90° | Less than a right angle | Roof pitch, ladder angles |
| Right | θ = 90° | Quarter turn | Building corners, L-brackets |
| Obtuse | 90° < θ < 180° | Greater than right angle | Some furniture joints |
| Straight | θ = 180° | Half turn, forms a line | Straight edges |
| Reflex | 180° < θ < 360° | Greater than straight angle | Clock hands, rotations |
| Complete | θ = 360° | Full rotation | Full circle, revolution |
Trigonometric Relationships Between Angles
Understanding how the six trigonometric functions relate to each other is essential for solving geometry and physics problems. These identities form the foundation of trigonometry as taught in UK GCSE and A-Level courses.
Fundamental Identities
The three primary trigonometric ratios -- sine, cosine and tangent -- are defined using the sides of a right-angled triangle. From any angle in a right triangle, the side opposite the angle, the side adjacent to it, and the hypotenuse determine all six functions:
- sin(theta) = opposite / hypotenuse -- measures the vertical component of the angle
- cos(theta) = adjacent / hypotenuse -- measures the horizontal component
- tan(theta) = opposite / adjacent = sin(theta) / cos(theta) -- the ratio of vertical to horizontal
The reciprocal functions are equally important in advanced mathematics:
- cosecant (csc) = 1 / sin(theta)
- secant (sec) = 1 / cos(theta)
- cotangent (cot) = 1 / tan(theta) = cos(theta) / sin(theta)
Key Pythagorean Identity
This identity holds true for every angle and is one of the most widely used relationships in mathematics. It derives directly from the Pythagorean theorem applied to the unit circle.
Complementary and Supplementary Angles in Detail
Two angles are complementary when they add up to exactly 90 degrees. This relationship creates an important connection between sine and cosine: the sine of any angle equals the cosine of its complement. For example, sin(30 degrees) = cos(60 degrees) = 0.5. This is why the co-functions carry the prefix "co" -- cosine literally means "complement's sine".
Two angles are supplementary when they add up to 180 degrees. Supplementary angles share the same sine value but have opposite cosine values. For instance, sin(150 degrees) = sin(30 degrees) = 0.5, but cos(150 degrees) = -cos(30 degrees). This symmetry is crucial when solving triangles using the law of sines.
Real-World Applications of Angles
Angles are not simply an abstract mathematical concept -- they have practical applications across many fields:
- Construction and Architecture: UK Building Regulations specify maximum stair pitch angles (typically 42 degrees for private stairs), roof pitch angles for different materials, and ramp gradients for accessibility compliance.
- Navigation and Aviation: Pilots and mariners measure bearings in degrees clockwise from true north. The bearing system is essential for flight paths, shipping routes around the UK coastline, and orienteering.
- Sports and Athletics: Launch angles determine projectile distance in football, cricket, and athletics. A 45-degree angle theoretically produces maximum range in vacuum, though air resistance modifies the optimum to roughly 40-42 degrees for most sports.
- Photography: The field of view of camera lenses is measured in degrees. Wide-angle lenses cover 64 to 84 degrees, while telephoto lenses narrow this to 10-30 degrees.
- Astronomy: Angular measurements describe positions of celestial objects. The full moon subtends approximately 0.5 degrees in the sky, and sextants measure angles between stars and the horizon for traditional navigation.
Full Rotation Angle Types
A full rotation (360 degrees or 2 pi radians) represents one complete turn. A reflex angle is greater than 180 degrees but less than 360 degrees. These angles appear frequently in engineering contexts such as measuring gear rotations, compass bearings beyond south, and the angular displacement of rotating machinery.
Practical Applications in the UK
Construction and Building
Angles are crucial in UK construction for roof pitches (typically 15-45°), stair calculations (UK Building Regulations specify 42° maximum pitch), and structural engineering.
Surveying and Land Measurement
UK surveyors commonly use both degrees and gradians. The Ordnance Survey uses precise angular measurements for mapping the entire country.
Navigation
Maritime and aviation navigation in the UK uses degrees for bearings. True north is 0°/360°, east is 90°, south is 180°, and west is 270°.
Education
Angles are taught throughout UK education from primary school (Key Stage 2) through GCSE and A-Level mathematics. Understanding angle conversions is essential for trigonometry and calculus.
UK Curriculum Relevance
This calculator supports GCSE and A-Level Mathematics curricula, covering angle measurement, trigonometric ratios, and radian measure as required by AQA, Edexcel, OCR, and other UK exam boards.
Frequently Asked Questions
To convert degrees to radians, multiply the degree value by π/180 (approximately 0.01745329). For example, 90° = 90 × (π/180) = π/2 radians ≈ 1.5708 radians. This conversion is essential for trigonometric calculations in mathematics and physics.
Complementary angles are two angles that add up to 90° (a right angle). Supplementary angles are two angles that add up to 180° (a straight line). For example, if one angle is 35°, its complement is 55° (35° + 55° = 90°) and its supplement is 145° (35° + 145° = 180°).
Radians provide a natural relationship between angle measure and arc length (arc = radius × angle in radians). This makes calculus formulas simpler: the derivative of sin(x) is cos(x) only when x is in radians. Radians eliminate the need for conversion factors in many mathematical formulas.
Gradians (also called gon or grade) are primarily used in surveying and civil engineering, particularly in continental Europe. With 400 gradians in a full circle, a right angle is exactly 100 gradians, making mental calculations easier for surveyors working with land plots and building layouts.
The reference angle is the acute angle (between 0° and 90°) that an angle makes with the x-axis. For angles 0-90°, the reference angle is the angle itself. For 90-180°, subtract from 180°. For 180-270°, subtract 180°. For 270-360°, subtract from 360°. This helps calculate trigonometric values for any angle.
Cotangent is the reciprocal of tangent. cot(θ) = 1/tan(θ) = cos(θ)/sin(θ) = adjacent/opposite. It's one of the six trigonometric functions and is used in advanced mathematics, physics, and engineering calculations.
UK roof pitch is typically expressed as an angle in degrees. Common pitches range from 15° (low pitch for modern roofs) to 45° (traditional steep pitch). Building Regulations require minimum pitches for different roofing materials: 15° for interlocking tiles, 35° for plain tiles, and 25° for slates.
A reflex angle is an angle greater than 180° but less than 360°. To find the reflex angle of any angle, subtract it from 360°. For example, the reflex angle of 60° is 300° (360° - 60°). Reflex angles are common in rotational mechanics and clock calculations.
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Last updated: February 2026 | Verified with latest UK rates
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Understanding Your Results
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- Accurate formulas - Based on official UK standards
- Clear explanations - Understand how results are derived
- 2025/26 updated - Using current rates and regulations
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