Angle Converter
This angle converter accepts a value in degrees and converts it to radians, gradians, arcminutes, and arcseconds. It serves mathematicians, engineers, surveyors, and astronomers who work with different angular measurement systems.
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Formula
radians = degrees × π / 180
A full circle is 360 degrees, which equals 2π radians. Therefore one degree equals π/180 radians (approximately 0.01745 rad). The gradian divides a full circle into 400 units, so one degree equals 10/9 gradians. One degree equals 60 arcminutes, and one arcminute equals 60 arcseconds — giving 3,600 arcseconds per degree.
How to use the Angle Converter
- 1
Enter your angle (degrees)
Enter the angle in degrees to convert to radians, gradians, and other units.
- 2
Read your results instantly
Results update in real time as you type.
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Angle Units in Mathematics and Engineering
Degrees are the most familiar angle unit, dividing a full circle into 360 equal parts. Radians are the SI unit for angles and are preferred in mathematics and physics because they simplify many formulas — for example, the arc length formula s = rθ only works cleanly in radians. Gradians (or grades) divide a full circle into 400 parts, making right angles exactly 100 gradians. This is convenient for surveying and civil engineering, where slope percentages often correspond directly to gradian values. Trigonometric functions in most programming languages accept radians by default.
Arcminutes and Arcseconds in Astronomy and Navigation
Arcminutes (′) and arcseconds (″) are subdivisions of degrees used in astronomy, geodesy, and navigation. One degree contains 60 arcminutes; one arcminute contains 60 arcseconds. The Moon's apparent diameter is about 30 arcminutes (0.5°). The angular resolution of the human eye is about 1 arcminute. In geodesy, one arcsecond of latitude corresponds to about 31 meters on Earth's surface. GPS coordinates use degrees, arcminutes, and arcseconds (DMS format) interchangeably with decimal degrees. Astronomical coordinates (right ascension and declination) use hours/minutes/seconds and degrees/arcminutes/arcseconds respectively.
Tips & Insights
Radians in Trigonometry
When using trigonometric functions (sin, cos, tan) in programming, always check whether the function expects degrees or radians. Most programming languages (Python, JavaScript, C) use radians by default. Convert with: radians = degrees × π / 180.
Quick Degree-to-Radian Shortcuts
Key radian values: 0° = 0 rad, 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π. Memorizing these makes angle calculations much faster without needing a calculator.
GPS Coordinates in DMS
GPS coordinates can be expressed in degrees-minutes-seconds (DMS) or decimal degrees (DD). To convert DMS to DD: DD = degrees + (minutes/60) + (seconds/3600). For example, 40° 26′ 46″ N = 40 + 26/60 + 46/3600 = 40.4461° N.
Worked Examples
Convert 90 Degrees (Right Angle)
90° equals π/2 ≈ 1.5708 radians, 100 gradians, 5,400 arcminutes, and 324,000 arcseconds.
Convert 180 Degrees (Straight Line)
180° equals π ≈ 3.14159 radians, 200 gradians, 10,800 arcminutes, and 648,000 arcseconds.
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Frequently Asked Questions
Why are there 360 degrees in a circle?
The Babylonians used a base-60 number system and divided the year into 360 days, assigning one degree to each day of solar travel. The number 360 is also highly composite (divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180), making it convenient for geometry.
What is a radian?
A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. Since a full circle's circumference is 2π times the radius, a full circle equals 2π radians, and one radian equals about 57.296 degrees.
What is a gradian?
A gradian (grad or gon) divides a full circle into 400 equal parts, making a right angle exactly 100 gradians. It is used in surveying, particularly in European engineering contexts, where slopes and gradients often correspond cleanly to gradian values.
How many arcseconds are in a degree?
One degree equals exactly 3,600 arcseconds (60 arcminutes × 60 arcseconds per arcminute). This fine subdivision is used in astronomy and geodesy where very small angular differences are significant.
How do I convert decimal degrees to degrees, minutes, and seconds?
Take the integer part as degrees. Multiply the decimal remainder by 60 to get minutes (take the integer part). Multiply the remaining decimal by 60 to get seconds. For example, 40.4461° → 40° 26.77′ → 40° 26′ 46.2″.
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