Degree (°) - Unit Information & Conversion

Quick Answer

What is a Degree? A degree (°) is a unit of angle equal to 1/360th of a full circle. Used universally for measuring angles in geometry, navigation, and everyday applications. 90° = right angle, 180° = straight angle, 360° = full circle. Use our angle converter to convert degrees to radians, gradians, and other units instantly.

Key Facts: Degree

Property	Value
Symbol	°
Quantity	Angle
System	Metric/SI Derived
Derived from	Radian
Category	Angle
Standard Body	NIST / ISO

Quick Comparison Table

Degrees	Radians	Gradians	Context	Convert Now
0°	0	0	Starting point	° to rad →
30°	π/6 ≈ 0.524	33.33	Common angle	° to rad →
45°	π/4 ≈ 0.785	50	Diagonal	° to rad →
60°	π/3 ≈ 1.047	66.67	Equilateral triangle	° to rad →
90°	π/2 ≈ 1.571	100	Right angle	° to rad →
180°	π ≈ 3.142	200	Straight angle	° to rad →
360°	2π ≈ 6.283	400	Full circle	° to rad →

Definition

A degree (°) divides a full circle into 360 equal parts. 1° = 1/360 of a full rotation = 60 arcminutes = 3600 arcseconds.

The degree is uniquely divisible:

• 360 = 2³ × 3² × 5
• Divisible by: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180
• This divisibility made the Babylonian base-60 system practical for navigation and astronomy

History: Why 360?

Ancient Origins (1800 BCE - Babylonia)

The Babylonians, around 1800 BCE, chose 360 degrees for several practical reasons:

• Sexagesimal System: Base-60 was their mathematical foundation (still seen in 60 seconds/minute, 60 minutes/hour)
• Solar Year Approximation: The year has roughly 365 days, and 360 was close enough for early calendars
• High Divisibility: 360 has 24 divisors—more than any smaller number except highly composite numbers
  • This made navigation with compass bearings straightforward
  • Allows easy division into halves (180°), quarters (90°), thirds (120°), sixths (60°), etc.
• Astronomical Observation: Ancient astronomers could divide the zodiac into 12 houses (each 30°)

Why Not 100?

The metric system later proposed 100 divisions (gradians), but degrees persisted because:

• Established Tradition: 2,000+ years of navigation, astronomy, and mathematics built on degrees
• Natural Divisibility: 360 divides more naturally than 100 for common angles
• Seamless Sub-division: 1° = 60' = 3600" (arcminutes and arcseconds) flow naturally from base-60
• Inertia in Navigation: Maritime and aviation industries deeply invested in degree-based instruments

Modern Standardization

• 1665: The degree symbol (°) officially adopted in European mathematics
• 1960s: Gradians promoted as "metric alternative" but never gained widespread adoption outside Europe
• Today: Degrees remain globally dominant; gradians used mainly in French/Swiss surveying

Degree Structure: Minutes and Seconds

The degree is subdivided using the sexagesimal (base-60) system:

Unit	Symbol	Equivalent	Decimal Degrees
Degree	°	1°	1.0
Arcminute	' (or m)	1° = 60'	1/60 = 0.0167
Arcsecond	" (or s)	1' = 60"	1/3600 = 0.000278

Example: 45° 30' 15" = 45.5042°

This system remains standard in navigation, surveying, and astronomy.

Common Uses

Navigation: Compass bearings

• North = 0° (or 360°)
• East = 90°
• South = 180°
• West = 270°

Geometry: Measuring angles in triangles, polygons, and circles

• Acute angle: 0-90°
• Right angle: 90°
• Obtuse angle: 90-180°
• Reflex angle: 180-360°

Geography: Latitude and longitude coordinates

• Latitude: 0° (Equator) to ±90° (Poles)
• Longitude: 0° (Prime Meridian) to ±180°

Construction: Slope angles, roof pitch, ramp grades

• Wheelchair ramp: 4.76° maximum (1:12 slope)
• Standard roof pitch: 20-30° (steep roofs 45°+)
• Stairs: 30-35° typical

Real-World Applications by Industry

Surveying and Land Measurement

• Bearing Notation: N 45°E means 45° clockwise from north (northeast)
• Declination: Magnetic north vs. true north (varies by location, ~10-20°)
• Topographic Maps: Contour lines use degree-based coordinates
• Property Lines: Lot boundaries specified in degrees (e.g., "bearing S 30°W for 200 feet")

Aviation and Navigation

• Heading: Direction aircraft travels (0° = North, 90° = East, etc.)
• Magnetic Compass: 360-degree dial fundamental to navigation
• Elevation: Climb angle measured in degrees (typical: 10-20°)
• Bank Angle: Aircraft tilt during turns (commercial flights: max 30-40°)
• Approach Angle: Glide slope for landing (typically 3° descent)

Maritime Navigation

• Course Over Ground (COG): Ship's track in degrees
• Heading: Direction ship's bow points
• Wind Direction: "Wind from the south" = wind traveling north (180°)
• Latitude Lines: Every 1° latitude ≈ 111 kilometers at sea
• Sextant Measurements: Traditionally measured in degrees/minutes/seconds

Astronomy and Space

• Right Ascension: Celestial coordinates in degrees/hours
• Declination: Celestial latitude from -90° (South Pole) to +90° (North Pole)
• Azimuth: Direction to celestial object (0° = North horizon, 90° = East horizon)
• Satellite Orbits: Inclination angle defines orbital plane relative to equator
  • Geostationary: 0° inclination (above equator)
  • Polar orbits: 90° inclination (over both poles)
• Solar Angles: Sun altitude affects shadows and solar panel efficiency

Physics and Engineering

• Velocity Vectors: Direction of motion specified in degrees
• Force Angles: Applied forces at specific angles to surfaces
• Projectile Motion: Launch angle determines range
  • 45° = maximum range in ideal conditions
  • 30° = commonly used for ballistics
• Light Diffraction: Angles measured in degrees (or smaller units)
• Mechanical Tolerances: Part angles specified to ±0.5° or tighter

Photography and Optics

• Field of View: Camera lens angle (50mm ≈ 50°, 24mm ≈ 80°)
• Angle of View: How wide the lens "sees" the scene
• Tilt: Camera angle relative to horizon
• Light Angle: Key light position affects shadow and mood

Degree Subdivisions: Minutes and Seconds

Context	Notation	Decimal	Example Use
Standard (degrees only)	45°	45.0°	Rough measurement, compass
Minutes precision	45° 30'	45.5°	Surveying, navigation
Seconds precision	45° 30' 15"	45.5042°	Geodesy, astronomy
Decimal degrees	45.5042°	45.5042°	GPS, digital systems

Conversion Between Formats

DMS to Decimal Degrees:

Decimal = Degrees + (Minutes / 60) + (Seconds / 3600)

Example: 45° 30' 15"
= 45 + (30/60) + (15/3600)
= 45 + 0.5 + 0.00417
= 45.50417°

Decimal Degrees to DMS:

1. Integer part = Degrees
2. Decimal × 60 = Minutes + remainder
3. Remainder × 60 = Seconds

Example: 45.50417°
= 45° + (0.50417 × 60)' + remainder''
= 45° 30' + (0.25 × 60)''
= 45° 30' 15"

Comparison with Other Angle Units

Quick Reference Table

Unit	Full Circle	Quadrant	Right Angle	Usefulness
Degrees	360°	90°	90°	⭐⭐⭐⭐⭐ Universal
Radians	2π	π/2	π/2	⭐⭐⭐⭐ Mathematical
Gradians	400ᵍ	100ᵍ	100ᵍ	⭐⭐ European surveying
Turns	1 turn	0.25 turn	0.25 turn	⭐⭐⭐ Rotations
Milliradians	2000π	500π	500π	⭐⭐ Military/sniping

When to Use Each Unit

• Degrees: Everyday use, navigation, construction, general angle measurement
• Radians: Calculus, trigonometry, physics, computer graphics
• Gradians: French/Swiss surveying, European civil engineering
• Turns: Describing rotations (e.g., "quarter turn" = 90°)
• Mils: Military targeting, precision optics

Precise Angle Measurement Instruments

Instrument	Precision	Used For
Compass	±5-10°	General direction
Protractor	±1°	Geometry, drawing
Surveyor's Level	±0.1°	Construction, grading
Theodolite	±0.01°	Land surveying
Total Station	±0.1-0.5" (seconds)	Professional surveying
Laser Theodolite	±0.3"	High-precision surveying
GPS/Coordinate System	±0.001°	Global positioning
Inclinometer	±0.5-1°	Slope measurement

Degrees in History: From Babylon to the Modern Era

The division of a circle into 360 degrees is one of the oldest and most enduring mathematical conventions in human history. This system traces its roots back to ancient Mesopotamia, specifically the Sumerians and later the Babylonians. The choice of 360 is not arbitrary; it is believed to be derived from the sexagesimal (base-60) number system used by these civilizations. 360 is a highly composite number, meaning it has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360). This makes it incredibly easy to divide a circle into equal parts for navigation, astronomy, and geometry.

The ancient astronomers noticed that the sun moved approximately one degree along the ecliptic each day, completing a full circuit in about 360 days (a standard ancient year). Thus, the degree became synonymous with a "step" in the celestial dance. Over millennia, this system was refined by the Greeks, including Hipparchus and Ptolemy, who used it to map the stars and the Earth. Today, despite the rise of the metric system and the radian, the 360-degree circle remains the standard for global navigation, aviation, and construction.

Degrees in Navigation: The Sextant and the Map

For centuries, the degree was the primary unit of navigation. Sailors used instruments like the sextant to measure the angle between a celestial body (like the Sun or the North Star) and the horizon. By knowing the exact degree of this angle and the time of day, navigators could determine their latitude on the Earth's surface.

In the modern era, GPS (Global Positioning System) uses degrees to define coordinates. Every point on Earth is mapped using Latitude (degrees North or South of the Equator) and Longitude (degrees East or West of the Prime Meridian). A single degree of latitude spans approximately 111 kilometers (69 miles), and global shipping and aviation depend on the precision of these degree-based measurements to cross oceans and continents safely.

Technical Benchmarks: Degrees in Precision Engineering

1. Machining Tolerance: High-precision CNC machines can resolve movements to /bin/zsh.001^{\circ}$.
2. Satellite Alignment: Orbital satellites must maintain their tilt within /bin/zsh.05^{\circ}$ to ensure signal stability.
3. Automotive Alignment: Car wheels are typically aligned with a "camber" angle within /bin/zsh.25^{\circ}$ of factory specifications.
4. Architectural Slope: ADA-compliant ramps must not exceed a slope of .76^{\circ}$ (1:12 ratio).
5. Optical Prisms: High-end binoculars use prisms ground to an accuracy of 1 arcsecond (1/3600th of a degree).
6. Solar Tracking: Photovoltaic arrays adjust their angle by ^{\circ}$ every hour to follow the sun's path.
7. Wind Turbine Pitch: Blades adjust their angle of attack in ^{\circ}$ increments to optimize energy capture.
8. Artillery Elevation: Modern howitzers calculate firing solutions with millidegree precision.
9. Display Viewing Angles: High-end IPS monitors maintain color accuracy up to ^{\circ}$.
10. Robot Arm Kinematics: Industrial robots move through 6 degrees of freedom with /bin/zsh.01^{\circ}$ repeatability.

The Global Degree Index: 200 Common Angles and Their Meaning

1. 0 Degrees: True North; The horizon line; The starting point.
2. 1 Degree: The daily path of the sun (approx); The width of a finger at arm's length.
3. 5 Degrees: The tilt of the Moon's orbit relative to the Earth's orbit.
4. 10 Degrees: The width of a fist at arm's length (used in star-gazing).
5. 15 Degrees: The rotation of the Earth in one hour; One timezone.
6. 23.5 Degrees: The axial tilt of the Earth (responsible for the seasons).
7. 30 Degrees: The angle of a standard drafting triangle.
8. 42 Degrees: The angle at which a rainbow appears relative to the observer.
9. 45 Degrees: A perfect diagonal; Half of a right angle.
10. 51.5 Degrees: The slope of the Great Pyramid of Giza.
11. 60 Degrees: The internal angle of an equilateral triangle.
12. 66.5 Degrees: The latitude of the Arctic and Antarctic Circles.
13. 90 Degrees: A right angle; Perpendicular; The "Normal" in physics.
14. 120 Degrees: The internal angle of a regular hexagon (honeycombs).
15. 135 Degrees: The angle of an octagon (Stop signs).
16. 180 Degrees: A straight line; A U-turn; Opposition in astronomy.
17. 270 Degrees: Due West (if starting from North).
18. 360 Degrees: A full circle; A "Full Monty"; Back where you started.