Angle Unit Converter

Convert between all angle units instantly. Our comprehensive angle converter handles degrees, radians, gradians, arcminutes, arcseconds, and all common angular measurements for mathematics, navigation, surveying, and astronomy.

Quick Angle Conversions

Most Popular Angle Conversions

• Degrees to Radians → - Math and trigonometry
• Radians to Degrees → - Convert from SI units
• Degrees to Gradians → - European surveying
• Arcminutes to Degrees → - Precision angles
• Arcseconds to Degrees → - Astronomy

By Application

Mathematics & Trigonometry:

• Degrees to Radians → - π radians = 180°
• Radians to Degrees → - Used in calculus

Navigation & Geography:

• Degrees to Arcminutes → - Latitude/longitude precision
• Arcminutes to Arcseconds → - GPS coordinates

Surveying & Engineering:

• Degrees to Gradians → - Continental Europe standard
• Gradians to Degrees → - 400 gradians = 360°

Astronomy:

• Arcseconds to Degrees → - Stellar positions
• Degrees to Arcseconds → - Angular diameter

Rotational Motion:

• Turns to Degrees → - Full rotations
• Revolutions to Radians → - RPM calculations

Understanding Angle Units

What is an Angle?

An angle measures the amount of rotation or turn between two lines or rays that share a common endpoint (vertex). Angles are fundamental in geometry, trigonometry, navigation, and engineering.

Key Concepts:

• Full rotation = 360° = 2π radians = 400 gradians = 1 turn
• Right angle = 90° = π/2 radians = 100 gradians
• Straight angle = 180° = π radians = 200 gradians

Common Angle Units Explained

Degree (°)

Most common angle unit. Divides a full circle into 360 equal parts.

Why 360?

• Ancient Babylonian mathematics (base 60)
• 360 has many divisors (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, etc.)
• Approximately matches days in a year

Common Uses:

• Geometry and basic math
• Navigation (compass bearings)
• Temperature (different meaning)
• Latitude and longitude
• Slope and grade percentages
• Protractors and angle measurement

Common Angles:

• Right angle: 90°
• Straight angle: 180°
• Acute angle: 0° to 90°
• Obtuse angle: 90° to 180°
• Full rotation: 360°

Convert Degrees →

Radian (rad)

SI unit of angle. Defined by the radius of a circle.

Definition: One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.

Key Values:

• π radians = 180°
• 2π radians = 360° (full circle)
• 1 radian ≈ 57.2958°
• π/2 radians = 90° (right angle)
• π/4 radians = 45°

Why Radians?

• Natural unit for calculus and physics
• Simplifies trigonometric formulas
• sin(x) ≈ x for small x (when x in radians)
• Arc length = radius × angle (in radians)

Common Uses:

• Advanced mathematics
• Physics and engineering
• Trigonometry calculations
• Calculus (derivatives, integrals)
• Angular velocity (rad/s)
• Wave equations

Convert Radians →

Gradian (grad) / Gon

Decimal angle unit. Divides a right angle into 100 equal parts.

Definition: 400 gradians = 360 degrees = full circle

Conversions:

• 1 gradian = 0.9°
• 100 gradians = 90° (right angle)
• 200 gradians = 180° (straight angle)
• 400 gradians = 360° (full circle)

Common Uses:

• Surveying in continental Europe
• Some calculators (GRAD mode)
• Topographic mapping
• Civil engineering

Advantages:

• Decimal-based (easier calculations)
• Right angle = exactly 100 gradians
• Works well with metric system

Convert Gradians →

Arcminute (′) and Arcsecond (″)

Subdivisions of degrees for high-precision measurements.

Arcminute (minute of arc):

• 1 arcminute = 1/60 degree
• 1° = 60 arcminutes
• Symbol: ′ (prime)

Arcsecond (second of arc):

• 1 arcsecond = 1/60 arcminute = 1/3600 degree
• 1° = 3,600 arcseconds
• Symbol: ″ (double prime)

Common Uses:

Navigation & Geography:

• GPS coordinates: 40°26′46″N, 79°58′56″W
• 1 arcminute ≈ 1 nautical mile (at equator)
• 1 arcsecond ≈ 30 meters (at equator)

Astronomy:

• Angular diameter of celestial objects
• Moon diameter: ~31 arcminutes
• Jupiter diameter: ~40 arcseconds
• Parallax measurements

Optics:

• Visual acuity (20/20 vision ≈ 1 arcminute resolution)
• Telescope resolution
• MOA (Minute of Angle) in shooting

Convert Arcminutes → | Convert Arcseconds →

Turn / Revolution

Complete rotation around a circle.

Definition:

• 1 turn = 360°
• 1 turn = 2π radians
• 1 turn = 400 gradians

Common Uses:

• Rotational speed (RPM = revolutions per minute)
• Mechanical engineering
• Spiral geometry
• Winding numbers in mathematics

Example: A car wheel turning at 600 RPM makes 10 revolutions per second.

Convert Turns →

Quadrant

Quarter of a circle = 90°.

Definition: 4 quadrants = 360° (full circle)

Common Uses:

• Coordinate plane geometry (Quadrant I, II, III, IV)
• Navigation sectors
• Historical angle measurement

Mil (Military Mil)

Military and artillery angle unit.

NATO Standard:

• 6,400 mils = 360°
• 1 mil ≈ 0.05625°
• 1,600 mils = 90° (right angle)

Other Standards:

• Swedish: 6,300 mils per circle
• Soviet: 6,000 mils per circle

Why Mils?

• Easy mental calculations in field
• 1 mil ≈ 1 meter at 1,000 meters distance
• Used for artillery range finding

Common Uses:

• Military operations
• Artillery fire control
• Sniper scope adjustments
• Compass bearings

Angle Conversions by Context

Geometry & Trigonometry

Common Angle	Degrees	Radians	Gradians
Zero	0°	0	0 grad
Acute	30°	π/6 ≈ 0.524	33.33 grad
Acute	45°	π/4 ≈ 0.785	50 grad
Acute	60°	π/3 ≈ 1.047	66.67 grad
Right angle	90°	π/2 ≈ 1.571	100 grad
Obtuse	120°	2π/3 ≈ 2.094	133.33 grad
Obtuse	135°	3π/4 ≈ 2.356	150 grad
Obtuse	150°	5π/6 ≈ 2.618	166.67 grad
Straight	180°	π ≈ 3.142	200 grad
Reflex	270°	3π/2 ≈ 4.712	300 grad
Full circle	360°	2π ≈ 6.283	400 grad

Convert between angle units →

Navigation & GPS Coordinates

Coordinate Format: Degrees, Minutes, Seconds (DMS)

Example: New York City

• 40° 42′ 46″ N, 74° 0′ 21″ W
• Decimal: 40.7128° N, 74.0060° W

Precision at Equator:

• 1 degree ≈ 111 km (69 miles)
• 1 arcminute ≈ 1.85 km (1.15 miles, ~1 nautical mile)
• 1 arcsecond ≈ 31 meters (102 feet)
• 0.1 arcsecond ≈ 3 meters (10 feet)
• 0.01 arcsecond ≈ 30 cm (1 foot)

Common Navigation Angles:

• North: 0° (or 360°)
• East: 90°
• South: 180°
• West: 270°
• Northeast: 45°
• Southeast: 135°
• Southwest: 225°
• Northwest: 315°

Convert GPS coordinates →

Astronomy & Celestial Objects

Angular Sizes (as seen from Earth):

Object	Angular Diameter
Sun	32 arcminutes (0.53°)
Moon	31 arcminutes (0.52°)
Venus (max)	1 arcminute (60 arcseconds)
Jupiter (max)	50 arcseconds
Mars (max)	25 arcseconds
Saturn (max)	20 arcseconds
Uranus	4 arcseconds
Neptune	2.5 arcseconds
Pluto	0.1 arcseconds
Andromeda Galaxy	3° × 1° (180′ × 60′)
Hubble Deep Field	2.6 arcminutes

Stellar Parallax:

• 1 parsec = distance where parallax = 1 arcsecond
• Proxima Centauri parallax: 0.77 arcseconds
• Most stars: < 0.05 arcseconds parallax

Convert arcseconds for astronomy →

Angular Velocity & Rotation

RPM Conversions:

Description	RPM	Degrees/sec	Radians/sec	Rev/sec
Slow motor	60 RPM	360°/s	6.28 rad/s	1 rev/s
Ceiling fan (low)	100 RPM	600°/s	10.5 rad/s	1.67 rev/s
Car engine (idle)	800 RPM	4,800°/s	83.8 rad/s	13.3 rev/s
Car wheel (60 mph)	850 RPM	5,100°/s	89.0 rad/s	14.2 rev/s
Washing machine (spin)	1,200 RPM	7,200°/s	125.7 rad/s	20 rev/s
Car engine (highway)	2,500 RPM	15,000°/s	261.8 rad/s	41.7 rev/s
Drill	3,000 RPM	18,000°/s	314.2 rad/s	50 rev/s
Jet engine	10,000+ RPM	60,000+°/s	1047+ rad/s	167+ rev/s

Formula: Angular velocity (rad/s) = RPM × 2π / 60

Real-World Examples

Example 1: GPS Coordinates Precision

Question: You have GPS coordinates 40°26′46″N. What is this in decimal degrees, and how precise is it?

Calculation:

• 40 degrees
• 26 arcminutes = 26/60 = 0.4333 degrees
• 46 arcseconds = 46/3600 = 0.0128 degrees
• Total: 40 + 0.4333 + 0.0128 = 40.4461°N

Precision: The arcsecond precision (46″) gives accuracy of about 30 meters (100 feet) at this latitude.

Convert GPS coordinates: DMS to Decimal →

Example 2: Trigonometry Calculation

Question: Calculate sin(30°) using radians.

Step 1: Convert 30° to radians

• 30° = 30 × (π/180) = π/6 ≈ 0.5236 radians

Step 2: Calculate

• sin(π/6) = sin(0.5236 rad) = 0.5

Why it matters: Many calculators and programming languages require angles in radians for trig functions.

Convert degrees to radians →

Example 3: Surveying with Gradians

Question: A surveyor measures a horizontal angle of 85 gradians. What is this in degrees?

Calculation:

• 1 gradian = 0.9 degrees
• 85 gradians × 0.9 = 76.5°

Context: Gradians are preferred in some European surveying because 100 gradians = exactly 90° (right angle), making calculations simpler.

Convert gradians →

Example 4: Telescope Resolution

Question: A telescope can resolve objects 1 arcsecond apart. Can it distinguish two stars separated by 0.5 arcminutes?

Calculation:

• 0.5 arcminutes = 0.5 × 60 = 30 arcseconds

Answer: Yes! The separation (30 arcseconds) is much larger than the telescope's resolution (1 arcsecond), so the two stars will appear clearly separated.

Example 5: Sniper Scope Adjustment

Question: A sniper scope adjusts in 1/4 MOA (minute of angle) clicks. How many clicks to adjust 2 inches at 100 yards?

Background:

• 1 MOA ≈ 1 inch at 100 yards
• Scope adjusts in 1/4 MOA = 0.25 inches per click

Calculation:

• Adjustment needed: 2 inches
• Clicks needed: 2 ÷ 0.25 = 8 clicks

Answer: 8 clicks on the scope adjustment.

Common Conversion Mistakes to Avoid

❌ Mistake #1: Calculator in Wrong Mode

Wrong: Calculating sin(30) and getting -0.988 instead of 0.5

Problem: Calculator is in radian mode, not degree mode

Correct:

• sin(30°) = 0.5 (calculator in DEG mode)
• sin(30 radians) = -0.988 (calculator in RAD mode)

Fix: Always check your calculator mode (DEG/RAD/GRAD) before trig calculations.

❌ Mistake #2: Forgetting π in Radian Conversions

Wrong: 180 radians = 180° ❌

Correct:

• 180° = π radians ≈ 3.14159 radians
• 180 radians = 10,313.24°

Formula:

• Degrees to radians: multiply by π/180
• Radians to degrees: multiply by 180/π

Convert correctly: Degrees to Radians →

❌ Mistake #3: Confusing Arcminutes with Minutes

Wrong: 30 arcminutes = 30 minutes of time ❌

Correct:

• Arcminutes (′) measure ANGLES
• Minutes measure TIME
• 30 arcminutes = 0.5 degrees
• 30 minutes = 1,800 seconds (time)

Context: In astronomy, both are used, but they're completely different units!

❌ Mistake #4: Wrong DMS Conversion

Wrong: 40.5° = 40°5′0″ ❌

Correct: 40.5° = 40°30′0″

• 0.5° = 0.5 × 60 = 30 arcminutes

Formula:

• Decimal degrees × 60 = minutes
• Remaining decimal × 60 = seconds

❌ Mistake #5: Negative Angle Direction

Wrong: -90° is the same as 90° ❌

Correct:

• Positive angles: counter-clockwise rotation
• Negative angles: clockwise rotation
• -90° = 270° (when normalized to 0-360°)
• But they represent different rotational directions!

Angle Conversion Formulas

Degrees ↔ Radians

Degrees to Radians:

• radians = degrees × (π/180)
• radians = degrees × 0.0174533

Radians to Degrees:

• degrees = radians × (180/π)
• degrees = radians × 57.2958

Examples:

• 90° = 90 × (π/180) = π/2 ≈ 1.571 radians
• 1 radian = 1 × (180/π) ≈ 57.296°

Degrees ↔ Gradians

Degrees to Gradians:

• gradians = degrees × (10/9)
• gradians = degrees × 1.1111

Gradians to Degrees:

• degrees = gradians × (9/10)
• degrees = gradians × 0.9

Examples:

• 90° = 90 × 1.1111 = 100 gradians
• 200 gradians = 200 × 0.9 = 180°

Degrees ↔ Arcminutes ↔ Arcseconds

Degrees to Arcminutes:

• arcminutes = degrees × 60

Degrees to Arcseconds:

• arcseconds = degrees × 3,600

DMS to Decimal Degrees:

• decimal = degrees + (minutes/60) + (seconds/3,600)

Examples:

• 1° = 60 arcminutes = 3,600 arcseconds
• 45°30′15″ = 45 + (30/60) + (15/3600) = 45.5042°

Turns and Revolutions

Turns to Degrees:

• degrees = turns × 360

Revolutions to Radians:

• radians = revolutions × 2π

Examples:

• 0.5 turns = 180°
• 2 revolutions = 4π radians ≈ 12.566 radians

Mils to Degrees

NATO Mil to Degrees:

• degrees = mils × (360/6400)
• degrees = mils × 0.05625

Degrees to NATO Mils:

• mils = degrees × (6400/360)
• mils = degrees × 17.778

Examples:

• 1,600 mils = 90°
• 45° = 800 mils

Quick Reference Table

From	To	Multiply by	Example
Degree	Radian	π/180 ≈ 0.01745	90° = 1.571 rad
Radian	Degree	180/π ≈ 57.296	1 rad = 57.3°
Degree	Gradian	10/9 ≈ 1.111	90° = 100 grad
Gradian	Degree	9/10 = 0.9	100 grad = 90°
Degree	Arcminute	60	1° = 60′
Arcminute	Degree	1/60 ≈ 0.01667	60′ = 1°
Degree	Arcsecond	3,600	1° = 3,600″
Arcsecond	Degree	1/3,600 ≈ 0.000278	3,600″ = 1°
Degree	Turn	1/360 ≈ 0.00278	360° = 1 turn
Turn	Degree	360	1 turn = 360°
Degree	Quadrant	1/90 ≈ 0.0111	90° = 1 quad
Degree	Mil (NATO)	6400/360 ≈ 17.78	90° = 1,600 mil

Use our calculator for precise conversions →

Angle FAQs

How do I convert degrees to radians?

Formula: radians = degrees × (π/180)

Examples:

• 30° = 30 × 0.01745 = 0.524 radians = π/6
• 45° = 45 × 0.01745 = 0.785 radians = π/4
• 60° = 60 × 0.01745 = 1.047 radians = π/3
• 90° = 90 × 0.01745 = 1.571 radians = π/2
• 180° = 180 × 0.01745 = 3.142 radians = π

Quick mental math:

• Divide degrees by 57.3 for rough estimate
• Remember key values: π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°), π (180°)

Use our Degrees to Radians converter →

Why do we use radians instead of degrees?

Mathematical Reasons:

1. Natural unit for calculus: d/dx[sin(x)] = cos(x) ONLY when x is in radians
2. Arc length formula: s = rθ (simple!) when θ is in radians
3. Small angle approximation: sin(x) ≈ x when x is in radians and small
4. Taylor series: Simpler formulas with radians

Physical Reasons:

• Angular velocity naturally expressed in rad/s
• Rotational kinetic energy formulas simpler
• Wave equations more elegant

When to use each:

• Radians: Physics, calculus, programming, engineering calculations
• Degrees: Navigation, surveying, everyday angles, communication

How accurate is GPS in arcminutes vs arcseconds?

Coordinate Precision at Equator:

• Degrees (e.g., 40°): ±111 km (69 miles) - Country level
• 0.1 degree: ±11 km (6.9 miles) - City level
• Arcminute (e.g., 40°26′): ±1.85 km (1.15 miles) - Neighborhood
• 0.1 arcminute (6″): ±185 meters (607 feet) - Block level
• Arcsecond (e.g., 40°26′46″): ±31 meters (102 feet) - Building level
• 0.1 arcsecond: ±3 meters (10 feet) - Room level
• 0.01 arcsecond: ±0.3 meters (1 foot) - Precision survey

Consumer GPS: Typically accurate to 3-10 meters (about 0.1-0.3 arcseconds)

Differential GPS: Can achieve centimeter-level accuracy

Convert GPS coordinates →

What is the difference between degrees and gradians?

Degrees:

• 360 degrees = full circle
• 90 degrees = right angle
• Based on Babylonian mathematics
• Universal standard

Gradians (Gons):

• 400 gradians = full circle
• 100 gradians = right angle
• Decimal-based, metric-friendly
• Used in some European surveying

Conversion:

• 1 gradian = 0.9 degrees
• 1 degree = 1.1111 gradians

Why gradians exist:

• Easier decimal calculations
• Right angle = exactly 100 (not 90)
• Better fits metric system philosophy

Reality: Degrees won, and gradians are rarely used today outside specific European surveying contexts.

Convert Gradians to Degrees →

How many degrees is π radians?

π radians = 180 degrees (exactly)

This is the fundamental relationship between radians and degrees.

Key radian values:

• π/6 radians = 30°
• π/4 radians = 45°
• π/3 radians = 60°
• π/2 radians = 90°
• π radians = 180°
• 2π radians = 360° (full circle)

Why π?

• A radian is defined by the radius
• Circumference = 2πr
• Half circumference (180°) = πr
• So 180° = π radians

Quick approximation: π ≈ 3.14159, so π radians ≈ 3.14 radians ≈ 180°

Convert Radians to Degrees →

What is MOA in shooting?

MOA = Minute of Angle = 1 arcminute = 1/60 degree

Approximation: 1 MOA ≈ 1 inch at 100 yards

Actual value: 1 MOA = 1.047 inches at 100 yards

Scaling:

• 1 MOA at 100 yards ≈ 1 inch
• 1 MOA at 200 yards ≈ 2 inches
• 1 MOA at 300 yards ≈ 3 inches
• 1 MOA at 500 yards ≈ 5 inches

Scope Adjustments:

• Most scopes adjust in 1/4 MOA clicks
• 1/4 MOA ≈ 0.25 inches at 100 yards
• 4 clicks = 1 inch adjustment at 100 yards

Accuracy Standards:

• 1 MOA group = Very accurate rifle
• 0.5 MOA group = Match-grade accuracy
• 0.25 MOA group = Benchrest competition level

How do you convert DMS to decimal degrees?

DMS Format: Degrees° Minutes′ Seconds″

Formula: Decimal = D + (M/60) + (S/3600)

Examples:

Example 1: 40°26′46″N

• Decimal = 40 + (26/60) + (46/3600)
• Decimal = 40 + 0.4333 + 0.0128
• Decimal = 40.4461°N

Example 2: 79°58′56″W

• Decimal = 79 + (58/60) + (56/3600)
• Decimal = 79 + 0.9667 + 0.0156
• Decimal = 79.9822°W

Reverse (Decimal to DMS):

1. Degrees = integer part (40)
2. Minutes = decimal part × 60 (0.4461 × 60 = 26.766′)
3. Seconds = minutes decimal × 60 (0.766 × 60 = 46″)

Convert DMS coordinates →

What angle unit do engineers use?

It depends on the field:

Mechanical Engineering:

• Degrees for drawings, angles, rotations
• Radians for angular velocity, dynamics calculations
• Gradians rarely used (except some European firms)

Civil Engineering & Surveying:

• Degrees, minutes, seconds (DMS) for land surveying (US)
• Gradians in some European countries
• Decimal degrees for GPS/GIS work

Electrical Engineering:

• Radians for phase angles, AC circuits
• Degrees for phasor diagrams (communication)
• Radians/second for frequency (ω = 2πf)

Aerospace Engineering:

• Degrees for orientation, flight paths
• Radians for dynamics, orbital mechanics
• Both used extensively

Software Engineering:

• Radians almost exclusively (programming languages default to radians)
• Math libraries expect radians

Best practice: Always specify the unit in documentation to avoid confusion!

Related Conversions

• Length Converter → - Distance measurements
• Area Converter → - Surface measurements
• Speed Converter → - Linear velocity
• Time Converter → - Temporal measurements
• Force Converter → - Torque calculations (coming soon)

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