Degree Converter
Convert angles between degrees, radians, gradians, arcminutes, arcseconds, and turns. Used in trigonometry, navigation, surveying, and CAD design. See also our Degrees to Radians and Degrees to DMS converters.
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Popular Degree Converters:
What is a Degree Converter?
A degree converter is a tool that converts between different units of angle measurement such as degrees, radians, gradians, and arcminutes. It is essential for trigonometry, navigation, engineering, surveying, and any application involving angular measurements.
History of Degree Measurement
The 360-degree circle originated from ancient Babylon around 2000 BCE, likely because 360 approximates the days in a year and is highly divisible. The radian was introduced in the 18th century and became standard in mathematics due to its natural relationship with the circle. The gradian was developed during the French Revolution, dividing a right angle into 100 parts.
About This Degree Converter
This degree converter supports 13 different angle units including degree, radian, gradian (gon), mil, arcminute, arcsecond, turn, circle, quadrant, sextant, sign, grad, and right angle. It serves mathematicians, engineers, surveyors, and navigators.
Understanding Angle Measurement and Conversion
Angles measure rotation or the opening between two lines meeting at a point. They appear everywhere: architecture (roof pitch), navigation (compass bearings), engineering (shaft rotation), and mathematics (trigonometry). The most familiar unit is the degree (°), where a full rotation equals 360°. But mathematics and physics often use radians, where a full rotation equals 2π radians. Other systems include gradians (400 per revolution) and arcminutes/arcseconds for precise angular measurement.
The degree system dates to ancient Babylon, whose base-60 number system gave us 360° in a circle (roughly one degree per day of the year). Radians arose from mathematics — they make calculus formulas cleaner because the derivative of sin(x) is cos(x) only when x is in radians. Gradians (also called gons) were invented during the French Revolution alongside the metric system, dividing a right angle into exactly 100 parts for easier surveying calculations.
Angle conversion is essential whenever you move between mathematical computation (which uses radians) and practical measurement (which typically uses degrees). Programmers encounter this constantly because most programming languages' trigonometric functions expect radians, but user interfaces display degrees.
How to Convert Between Angle Units (Step-by-Step)
Angle conversion centers on the relationship between degrees and radians. Once you know that relationship, all other conversions follow logically.
- Know the fundamental relationship: 360° = 2π radians = 400 gradians (one full rotation).
- To convert degrees to radians: multiply by π/180. Example: 90° × (π/180) = π/2 radians.
- To convert radians to degrees: multiply by 180/π. Example: π/4 × (180/π) = 45°.
- To convert degrees to gradians: multiply by 10/9. Example: 90° × (10/9) = 100 gradians.
- For DMS (degrees-minutes-seconds) to decimal degrees: degrees + minutes/60 + seconds/3600. Example: 45°30'15" = 45 + 30/60 + 15/3600 = 45.504°.
Essential Angle Conversion Formulas
The key relationships between angle measurement systems:
- 1 full rotation = 360° = 2π radians = 400 gradians
- 1 degree = π/180 radians ≈ 0.017453 radians
- 1 radian = 180/π degrees ≈ 57.2958°
- 1 gradian = 0.9° = π/200 radians
- 1 degree = 60 arcminutes (60')
- 1 arcminute = 60 arcseconds (60")
- 1 degree = 3,600 arcseconds
- 1 radian = 3,437.75 arcminutes
Worked Examples — Angle Conversions
Example 1: Convert 135° to radians.
Solution:
Formula: radians = degrees × (π/180).
135 × (π/180) = 135π/180 = 3π/4 radians.
As a decimal: 3 × 3.14159/4 = 2.356 radians.
Answer: 135° = 3π/4 radians ≈ 2.356 radians.
Example 2: A GPS coordinate shows latitude 40°26'46"N. Convert to decimal degrees.
Solution:
Degrees: 40.
Minutes to degrees: 26 ÷ 60 = 0.4333°.
Seconds to degrees: 46 ÷ 3600 = 0.01278°.
Total: 40 + 0.4333 + 0.01278 = 40.4461°.
Answer: 40°26'46"N = 40.4461°N (approximately the latitude of New York City).
Example 3: A surveyor measures an angle of 150 gradians. Express in degrees and radians.
Solution:
Gradians to degrees: 150 × (9/10) = 135°.
Degrees to radians: 135 × (π/180) = 3π/4 ≈ 2.356 radians.
Answer: 150 gradians = 135° = 3π/4 radians.
Example 4: In a programming function, you need sin(30°) but the language uses radians. What value do you pass?
Solution:
Convert 30° to radians: 30 × (π/180) = π/6.
π/6 ≈ 0.5236 radians.
sin(0.5236) = 0.5 (confirming the conversion is correct).
Answer: Pass 0.5236 (π/6) radians to get sin(30°) = 0.5.
Example 5: The angular diameter of the Moon is about 31 arcminutes. Express in degrees and radians.
Solution:
Arcminutes to degrees: 31 ÷ 60 = 0.5167°.
Degrees to radians: 0.5167 × (π/180) = 0.009018 radians.
Answer: The Moon's angular diameter ≈ 0.517° ≈ 0.00902 radians (about half a degree).
Quick Reference — Common Angle Conversions
Frequently used angle equivalencies for mathematics, engineering, and navigation.
| From | To |
|---|---|
| 0° | 0 radians |
| 30° | π/6 rad ≈ 0.5236 |
| 45° | π/4 rad ≈ 0.7854 |
| 60° | π/3 rad ≈ 1.0472 |
| 90° | π/2 rad ≈ 1.5708 |
| 120° | 2π/3 rad ≈ 2.0944 |
| 180° | π rad ≈ 3.1416 |
| 270° | 3π/2 rad ≈ 4.7124 |
| 360° | 2π rad ≈ 6.2832 |
| 1 radian | 57.296° |
| 1 gradian | 0.9° |
| 1 arcminute | 1/60° |
| 1 arcsecond | 1/3600° |
| 1 milliradian | 0.0573° |
Degrees, Radians, and Gradians: Three Systems Compared
The degree is the oldest and most intuitive angle unit. Its 360-part division of a circle likely comes from the Babylonian approximation of the year (360 days) and their base-60 number system. The number 360 has many divisors (2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20...), making it easy to divide a circle into common fractions. This is why degrees remain dominant in navigation, architecture, and everyday geometry.
The radian is the natural mathematical unit of angle. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius. This definition connects angle directly to arc length: arc length = radius × angle (in radians). This simplicity makes radians essential for calculus, physics, and engineering. The derivative of sin(θ) is cos(θ) only in radians; in degrees, it would be (π/180)cos(θ) — an ugly constant that disappears with radians.
The gradian (also gon or grade) divides a right angle into exactly 100 parts, making a full circle 400 gradians. This was designed for surveying: slope percentages align naturally with gradians. A 1% slope corresponds to roughly 1 gradian. Though never widely adopted outside continental European surveying, gradians appear on most scientific calculators (the "GRAD" mode) and in some CAD software.
Where Angle Conversion Matters
Programming & Software
Nearly all programming languages (JavaScript, Python, C, Java) use radians for trigonometric functions. Any application displaying angles in degrees must convert: Math.sin(degrees * Math.PI / 180). Getting this wrong produces subtle, hard-to-debug errors.
Navigation & GPS
GPS coordinates use decimal degrees or DMS (degrees-minutes-seconds). Converting between these formats is essential for mapping applications. One degree of latitude ≈ 111 km, one arcminute ≈ 1.85 km (one nautical mile).
Astronomy
Star positions use hours-minutes-seconds for right ascension and degrees-arcminutes-arcseconds for declination. Telescope pointing requires converting to decimal degrees or radians for calculations.
Construction & Architecture
Roof pitch, stair angles, and structural slopes are specified in degrees. Carpenters convert between degree measurements and rise/run ratios constantly. A 45° angle has a 1:1 rise-to-run ratio.
Surveying & Mapping
Land surveyors measure angles in degrees (most countries) or gradians (France, some European countries). Bearing calculations require converting between true north (degrees) and grid north (may use gradians in some coordinate systems).
Physics & Engineering
Angular velocity (rad/s), rotational mechanics, and wave equations all require radians. An engineer who needs a motor spinning at 3,000 RPM must convert to rad/s: 3000 × 2π/60 = 314.16 rad/s.
Common Mistakes in Angle Conversion
The number one mistake is forgetting to convert degrees to radians before using trigonometric functions in code. Math.sin(90) in JavaScript does NOT give 1 — it gives sin(90 radians) = 0.894. You need Math.sin(90 * Math.PI / 180) = 1. Another common error is confusing arcminutes (1/60°) with decimal minutes. GPS coordinate 40°26.77' means 40 degrees and 26.77 minutes = 40.446°, NOT 40°26'77". The second-most common error is mixing up the direction of conversion: to go from larger units (degrees) to smaller numeric values (radians for angles under ~57°), you multiply by a factor less than 1 (π/180 ≈ 0.01745).
Why Radians Are "Natural" Units
Radians are considered the natural angle unit because they make many mathematical formulas simpler. The Taylor series for sine is: sin(x) = x - x³/3! + x⁵/5! - ... (only valid for x in radians). The small angle approximation sin(θ) ≈ θ works only in radians. The relationship between angular velocity, linear velocity, and radius (v = ωr) requires ω in radians per second. Electric AC signals described as sin(2πft) use radians implicitly. In essence, radians remove artificial scaling constants from the fundamental equations of mathematics and physics.
Key Takeaways
- One full rotation = 360° = 2π radians = 400 gradians.
- Degrees to radians: multiply by π/180 ≈ 0.01745.
- Radians to degrees: multiply by 180/π ≈ 57.296.
- Programming trig functions use radians — always convert if your input is in degrees.
- DMS to decimal: degrees + minutes/60 + seconds/3600.
- Key reference: 1 radian ≈ 57.3°, meaning a radian is a fairly large angle.
Metric Conversion Factor Tables for Degree Converter
| Units to convert | Multiply By The Number | Convert as Unit |
|---|---|---|
| Degree (°) | 0.0174532925 | Radian (rad) |
| Degree (°) | 1.1111111111 | Gradian (gon) |
| Degree (°) | 60 | Minute (') |
| Degree (°) | 3600 | Second (") |
| Degree (°) | 0.0027777778 | Turn |
| Radian (rad) | 57.295779513 | Degree (°) |
| Radian (rad) | 63.661977237 | Gradian (gon) |
| Radian (rad) | 0.1591549431 | Turn |
| Gradian (gon) | 0.9 | Degree (°) |
| Gradian (gon) | 0.0157079633 | Radian (rad) |
| Turn | 360 | Degree (°) |
| Turn | 6.2831853072 | Radian (rad) |
| Turn | 400 | Gradian (gon) |
| Quadrant | 90 | Degree (°) |
| Sextant | 60 | Degree (°) |
| Mil | 0.05625 | Degree (°) |
Degreeconverters & it's abbreviations
| Unit | Abbreviation | Unit | Abbreviation | Unit | Abbreviation |
|---|---|---|---|---|---|
| degree | ° | radian | rad | gradian | gon grad |
| mil | mil | minute of arc | ' arcmin | second of arc | " arcsec |
| turn | tr rev | circle | - | quadrant | - |
| sextant | - | sign | - | right angle | - |
| grad | ^g |
Frequently Asked Questions
How do I convert degrees to radians?
Multiply degrees by π/180 (approximately 0.0174533). For example, 90° = 90 × π/180 = π/2 ≈ 1.5708 radians.
How do I convert radians to degrees?
Multiply radians by 180/π (approximately 57.2958). For example, 1 radian = 1 × 180/π ≈ 57.296 degrees.
What is a gradian?
A gradian (also called gon or grad) divides a right angle into 100 parts, so a full circle is 400 gradians. It is used in surveying because it makes percentage slope calculations easier.
How many radians are in a full circle?
A full circle contains 2π radians (approximately 6.2832 radians). This equals 360 degrees, 400 gradians, or 1 turn.
What is an arcminute and arcsecond?
An arcminute is 1/60 of a degree. An arcsecond is 1/60 of an arcminute (1/3,600 of a degree). They are used in astronomy, navigation, and precise angular measurements.
Complete list of Degree conversion units and its conversion.
- 1 degree [°] = 0.0174532925 radian [rad]
degrees to radians → - 1 radian [rad] = 57.295779513 degree [°]
radians to degrees → - 1 degree [°] = 1.1111111111 gradian [gon]
degrees to gradians →
- 1 degree [°] = 0.0027777778 turn
degrees to turns → - 1 radian [rad] = 63.661977237 gradian [gon]
radians to gradians → - 1 radian [rad] = 3437.7467708 minute of arc [arcmin]
radians to arcminutes →
- 1 turn = 6.2831853072 radian [rad]
turns to radians → - 1 turn = 400 gradian [gon]
turns to gradians → - 1 circle = 360 degree [°]
circles to degrees →
- 1 sextant = 60 degree [°]
sextants to degrees → - 1 sextant = 1.0471975512 radian [rad]
sextants to radians → - 1 sign = 30 degree [°]
signs to degrees →
- 1 right angle = 1.5707963268 radian [rad]
right angles to radians → - 1 gradian [gon] = 0.0157079633 radian [rad]
gradians to radians →
- 1 gradian [gon] = 0.9 degree [°]
gradians to degrees → - 1 degree [°] = 60 minute of arc [arcmin]
degrees to arcminutes → - 1 degree [°] = 3600 second of arc [arcsec]
degrees to arcseconds →
- 1 radian [rad] = 206264.80625 second of arc [arcsec]
radians to arcseconds → - 1 radian [rad] = 0.1591549431 turn
radians to turns → - 1 turn = 360 degree [°]
turns to degrees →
- 1 circle = 6.2831853072 radian [rad]
circles to radians → - 1 quadrant = 90 degree [°]
quadrants to degrees → - 1 quadrant = 1.5707963268 radian [rad]
quadrants to radians →
- 1 mil = 0.05625 degree [°]
mils to degrees → - 1 mil = 0.0009817477 radian [rad]
mils to radians → - 1 right angle = 90 degree [°]
right angles to degrees →