Angle Converter
Convert between angle units including degrees, radians, gradians, and more.
90 deg =
1.570796
Radians (rad)
90 deg in all units
Quick Reference
90Β°
= Ο/2 rad β 1.5708
180Β°
= Ο rad β 3.1416
360Β°
= 2Ο rad β 6.2832
1 radian
β 57.2958Β°
What is Angle Conversion?
Angle conversion involves translating angular measurements between different unit systems. Angles are fundamental in mathematics, physics, engineering, navigation, and computer graphics. Different fields and regions prefer different angular units, making conversion between them an everyday necessity.
The degree is the most familiar angle unit, where a full circle equals 360 degrees. This convention dates back to ancient Babylonian astronomy. Radians are the SI derived unit for angles and are preferred in mathematics and physics because they simplify many formulas. A full circle equals 2Ο radians. Gradians (or grads) divide a right angle into 100 parts and are used in some surveying and civil engineering applications, particularly in Europe.
Beyond these common units, specialized fields use arc minutes and arc seconds for extremely precise angular measurements β astronomy, geodesy, and optics rely on these subdivisions. Milliradians are used in military applications for aiming and in precision engineering. This converter handles all these units and more, providing instant, accurate conversions.
Angle Conversion Formulas
Angle conversions use fixed ratios between units. The degree serves as the reference unit in this calculator, with each alternative unit defined by its relationship to one degree.
Degree to Radian Conversion
Where:
- degrees= Angle in degrees (Β°)
- radians= Angle in radians (rad)
- Ο= Pi, approximately 3.14159
Understanding Angle Units
Each angular unit has specific advantages and common use cases across different disciplines.
- Degrees (Β°): The most widely recognized unit. A full circle = 360Β°. Used in everyday navigation, geometry, and construction.
- Radians (rad): The mathematical standard. A full circle = 2Ο rad β 6.2832 rad. Essential for calculus and trigonometric functions in programming.
- Gradians (grad): A full circle = 400 grad. Used in French surveying and some European engineering applications.
- Turns: Represents complete rotations. One full rotation = 1 turn. Used in mechanical engineering and rotational systems.
- Arc minutes ('): 1/60 of a degree. Used in astronomy, navigation, and geodesy for high-precision measurements.
- Arc seconds ("): 1/60 of an arc minute, or 1/3600 of a degree. Used in astronomy and satellite positioning.
- Milliradians (mrad): 1/1000 of a radian. Used in military targeting and precision optics.
How to Use This Calculator
Converting angles between units is straightforward:
- Enter the angle value: Type the numerical value into the input field. For example, enter 90 for a right angle.
- Select the source unit: Choose the unit you are converting from using the "From" dropdown β degrees, radians, gradians, turns, arc minutes, arc seconds, or milliradians.
- Select the target unit: Choose the unit you want to convert to using the "To" dropdown.
- Read the result: The converted value appears instantly. You can also swap units with the swap button for quick reverse conversions.
- View all conversions: The all-units panel shows your input value converted to every available unit simultaneously, making comparison easy.
Real-World Applications
In navigation, bearings are expressed in degrees from 0Β° to 360Β°, measured clockwise from north. Pilots, mariners, and GPS systems all use degree-based bearings. Converting between degrees and other units is essential when interfacing with mathematical models or surveying equipment.
Computer graphics and game development rely heavily on radian-based angle calculations. Most programming languages' trigonometric functions (sin, cos, tan) expect radian inputs. Developers converting from design specifications in degrees must convert to radians before applying these functions.
Surveying and civil engineering use both degrees and gradians. The theodolite, a precision instrument for measuring angles, can typically display readings in either unit. Bridge alignments, road grades, and building angles all require precise angular measurements and frequent unit conversions.
Worked Examples
Degrees to Radians
Problem:
Convert 90 degrees to radians for use in a trigonometric calculation.
Solution Steps:
- 1Identify the conversion formula: radians = degrees Γ (Ο / 180)
- 2Substitute 90 degrees: radians = 90 Γ (Ο / 180)
- 3Simplify: radians = Ο / 2
- 4Calculate: Ο / 2 β 1.5708 radians
Result:
90 degrees equals Ο/2 radians or approximately 1.5708 rad
Radians to Degrees
Problem:
A wheel rotates 2Ο radians. How many degrees has it turned?
Solution Steps:
- 1Identify the conversion formula: degrees = radians Γ (180 / Ο)
- 2Substitute 2Ο: degrees = 2Ο Γ (180 / Ο)
- 3Simplify: degrees = 2 Γ 180 = 360
Result:
2Ο radians equals exactly 360 degrees (one full rotation)
Gradians to Degrees
Problem:
A surveyor measures an angle of 100 gradians. Convert this to degrees.
Solution Steps:
- 1Identify the conversion factor: 1 gradian = 0.9 degrees
- 2Multiply: 100 Γ 0.9 = 90 degrees
Result:
100 gradians equals exactly 90 degrees (a right angle)
Tips & Best Practices
- βRemember: Ο radians = 180 degrees. This is the fundamental conversion relationship.
- βUse radians for all programming trigonometric functions β degrees will give wrong results.
- βA right angle is 90Β°, Ο/2 radians, or 100 gradians β know these key reference points.
- βFor quick estimates: 1 radian β 57.3 degrees, and 1 degree β 0.0175 radians.
- βArc seconds are used for the highest precision β there are 3,600 arc seconds in one degree.
- βGradians are convenient because a right angle equals exactly 100 grad, simplifying field work.
Frequently Asked Questions
Sources & References
Last updated: 2026-06-06
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Editorial Note
MyCalcBuddy Editorial Team
This page is maintained as an educational calculator reference.
Formula Source: NIST Guide to SI Units
by National Institute of Standards