Home›Converters›Measurement›Angle Converter

Angle Converter

Measurement

Convert angle units instantly — degrees, radians, gradians, arcminutes, arcseconds, turns, and milliradians. Essential for maths, navigation, and optics.

Reviewed by the thecalcu.com team · Last updated July 1, 2026

From
To
All conversionsfor 1 Degrees (°)
Degrees (°)1
Radians (rad)0.017453293
Milliradians (mrad)17.453293
Gradians / Gon (grad)1.1111111
Arcminutes (′)60
Arcseconds (″)3600
Turns / Revolutions (τ)0.0027777778

What is a Angle?

An Angle Converter translates angle measurements between the unit systems used in mathematics, surveying, astronomy, navigation, optics, and everyday geometry, degrees, radians, milliradians, gradians, arcminutes, arcseconds, and turns.

An angle describes the rotation between two rays sharing a common endpoint. Despite measuring the same geometric quantity, different disciplines have settled on incompatible units based on the arithmetic that is most natural to their work.

Degrees (360 per full circle) dominate everyday usage because the Babylonian base-60 number system made 360 divisible by many factors. Radians (2π per full circle) are the mathematical standard because they make calculus identities clean, the derivative of sin(x) is cos(x) only when x is in radians. Gradians (400 per full circle) give surveyors a right angle of exactly 100 units for clean decimal arithmetic. Arcminutes and arcseconds subdivide the degree for astronomy and GPS precision. Milliradians are the preferred unit in precision optics and ballistics for intuitive distance-subtension arithmetic.

In India, angle units appear across many practical contexts. Indian school mathematics (CBSE/ICSE) teaches degrees through Class 10, then introduces radians in Class 11–12 for calculus and trigonometry. Engineering and physics courses use radians exclusively. Indian survey documents, particularly older land records, use degrees, minutes, and seconds (DMS format). Solar panel design uses degrees of inclination. ISRO satellite positioning uses arcminutes and arcseconds in orbital mechanics.

This converter handles the full range, including milliradians for technical users and turns for programmers working with rotation-based animation APIs. For rotational rates (how fast an angle changes), see the Frequency Converter.

Why Use an Angle Converter?

The dominant pain point is the degree-radian mismatch between everyday thinking and programming/mathematics.

  • Programming: A developer rotating a CSS element, animating a Three.js scene, or calling Python's math.sin() must convert degrees to radians. Forgetting the conversion is among the most common trigonometry bugs in beginner code.
  • GPS coordinates: Indian land documents and court records often list coordinates in DMS (degrees, minutes, seconds). Google Maps and GIS software use decimal degrees. Converting between them requires the arcminute and arcsecond relationships.
  • Surveying: Survey instruments calibrated in gradians produce grad readings that must be converted to degrees for CAD software input.
  • Scope adjustment: Long-range shooters and precision optics users adjusting in milliradians need to cross-reference with degree-based ballistic charts.

Who Should Use This Converter?

Mathematics and engineering students converting between degrees and radians for trigonometry problems, calculus, and physics applications in CBSE/JEE/GATE preparation.

Software developers and game developers building rotation logic, compass UIs, or physics engines that require radian inputs while the domain logic operates in degrees.

Surveyors and civil engineers using Total Station instruments that output gradians, needing conversion to degrees for AutoCAD or IS specification documentation.

Astronomers and astrophysics students working with star catalogue coordinates in arcminutes and arcseconds (RA/Dec format) and needing to cross-reference with decimal degrees for observation planning.

Precision shooters and optics technicians adjusting scope reticles in milliradians and referencing ballistic data in degrees of elevation. Pair with the Length Converter for distance-subtension calculations.

What Insights Does the Angle Converter Give You?

The primary output shows full decimal precision, 45° converts to exactly 0.78539816339 rad (= π/4), not a rounded 0.785.

The reference table displays all 7 units for your entered angle. Entering 180° immediately shows: 3.14159 rad, 3,141.59 mrad, 200 grad, 10,800 arcminutes, 648,000 arcseconds, 0.5 turns, a complete cross-reference that confirms the mathematical relationships.

Key angle landmarks across units:

Degrees Radians Gradians Turns
30° π/6 ≈ 0.5236 33.33 1/12
45° π/4 ≈ 0.7854 50 1/8
90° π/2 ≈ 1.5708 100 1/4
180° π ≈ 3.1416 200 1/2
360° 2π ≈ 6.2832 400 1

How to use this Angle calculator

  1. Select your source unit from the FROM dropdown, e.g. Degrees (°).
  2. Enter your value, e.g. 45 for a 45° angle.
  3. Select your target unit from the TO dropdown, e.g. Radians (rad).
  4. The result appears instantly, 45° = 0.7854 rad.
  5. Use ⇅ to swap FROM and TO directions.
  6. View the reference table to see your angle in all 7 units at once.
  7. Copy the URL to share a specific conversion.

Formula & Methodology

All units convert to and from degrees using exact multipliers:

| Unit | Symbol | Degrees (toBase) |
|---|---|---|
| Degree | ° | 1 (base) |
| Radian | rad | 180/π = 57.29577951… |
| Milliradian | mrad | 180/(π × 1000) = 0.05729578… |
| Gradian | grad | 0.9 (exact: 360/400) |
| Arcminute | ′ | 1/60 = 0.01666… |
| Arcsecond | ″ | 1/3600 = 0.000277… |
| Turn | τ | 360 (exact) |

Conversion formula: Result = Input × (F_from ÷ F_to)

Worked example, programming rotation:

A UI element needs to be rotated by 270° clockwise. The CSS rotate() function expects degrees, but a custom animation library uses radians.

270° × (π/180) = 270 × 0.017453293 = 4.71238898 rad

Pass 4.71239 to the animation function.

Common angle reference table:

| Angle | Degrees | Radians | Gradians |
|---|---|---|---|
| Right angle | 90° | π/2 = 1.5708 | 100 |
| Equilateral triangle | 60° | π/3 = 1.0472 | 66.67 |
| Isosceles right triangle | 45° | π/4 = 0.7854 | 50 |
| Full circle | 360° | 2π = 6.2832 | 400 |

Frequently Asked Questions

An angle converter translates an angle measurement between different unit systems, such as from degrees to radians, from gradians to arcminutes, or from turns to degrees. Angle units vary by discipline: degrees are used in everyday geometry and navigation; radians are the SI standard used in mathematics and physics; gradians (gon) are used in surveying; arcminutes and arcseconds are used in astronomy and GPS coordinates; milliradians are used in optics and ballistics.
Multiply degrees by π/180 (approximately 0.017453) to get radians. For example, 90° = 90 × π/180 = π/2 ≈ 1.5708 rad. Conversely, to convert radians to degrees, multiply by 180/π (approximately 57.2958). This is the most common angle conversion in mathematics and programming, since most trigonometric functions (sin, cos, tan) in scientific calculators and programming languages (Python, JavaScript, C) expect inputs in radians.
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of that circle. Since the circumference of a circle is 2πr, a full circle spans 2π radians = 360°. Radians are dimensionless (they are a ratio of two lengths) and are the natural unit for calculus involving trigonometric functions, derivatives of sin and cos are simple only when the angle is in radians.
A gradian (also called gon or grade) divides a full circle into 400 equal parts, so a right angle is exactly 100 gradians. This makes decimal arithmetic on right angles trivially clean, for surveyors calculating angles in the field, 153.7 grad is unambiguous without needing to convert minutes and seconds. Gradians are used in land surveying, civil engineering, and road design in continental Europe. Indian survey equipment following European standards may display grad readings.
An arcminute (′) is 1/60 of a degree, and an arcsecond (″) is 1/60 of an arcminute = 1/3600 of a degree. These fine subdivisions are used in astronomy (star positions, angular diameter of celestial objects), GPS coordinates (latitude and longitude in degrees, minutes, seconds, DMS format), and precision optics. The Moon's angular diameter is about 31 arcminutes (0.52°). GPS resolution to 1 arcsecond corresponds to roughly 31 metres on Earth's surface.
A milliradian (mrad) is 1/1000 of a radian ≈ 0.05730°. It is used in long-range shooting and ballistics because 1 mrad subtends approximately 1 cm at 10 m distance, or 10 cm at 100 m, making holdover adjustments on scope reticles intuitive. Milliradians are also used in military rangefinding, artillery, radar antenna beamwidths, and precision optical alignment. Riflescope turret adjustments are commonly specified in 0.1 mrad (= 1 cm at 100 m) clicks.
Take the decimal degree value, extract the whole number as degrees. Multiply the fractional part by 60, the whole number is arcminutes. Multiply the remaining fraction by 60 again, this gives arcseconds. For example, 28.6139°N (Delhi latitude): 28° whole degrees; 0.6139 × 60 = 36.834 arcminutes; 0.834 × 60 = 50.04 arcseconds → 28°36′50″N. Many Indian GPS and land survey documents still use the DMS format, particularly for plot coordinates in older records.
A full circle is 360 degrees, which equals 2π radians, 400 gradians, or 1 turn. Key reference angles: 90° = π/2 rad = 100 grad (right angle); 180° = π rad = 200 grad (straight line); 45° = π/4 rad = 50 grad; 30° = π/6 rad = 33.33 grad; 60° = π/3 rad = 66.67 grad. These are the angles of the standard triangles (45-45-90 and 30-60-90) that appear throughout Indian secondary school mathematics (CBSE Class 10 and 11 trigonometry).
Select your source unit from the FROM dropdown, for example, 'Degrees (°)'. Enter your value, such as 45 for a 45° angle. Choose your target unit from the TO dropdown, such as 'Radians (rad)'. The result appears instantly: 45° = 0.7854 rad (= π/4). Use the ⇅ swap button to reverse, and view the reference table below to see 45° expressed in all 7 angle units simultaneously.
Mathematical functions like sin, cos, and tan have simpler derivatives when angles are in radians, d/dx sin(x) = cos(x) only holds when x is in radians. Degrees would introduce a scaling constant of π/180 in every derivative. For this reason, Python's math.sin(), JavaScript's Math.sin(), C's sin(), and virtually all scientific computing libraries take radians. Developers must convert degrees to radians before passing to these functions: radians = degrees × Math.PI / 180.
The human eye can resolve details separated by approximately 1 arcminute (1′ = 1/60°) under ideal conditions. This is why standard visual acuity (20/20 vision) is defined as the ability to distinguish letters subtending 5 arcminutes. Screen resolution is designed around this: at a comfortable viewing distance of 30 cm, a display needs about 300 pixels per inch (PPI) to appear 'retina quality', pixels smaller than 1 arcminute in apparent size.
Indian construction uses degrees for roof slopes, staircase angles, and ramp gradients. A standard staircase tread-to-riser ratio in NBC 2016 gives an angle of approximately 26–35°. Solar panel inclination angles for India range from 10° (near the equator in Tamil Nadu) to 32° (in Himalayan states), typically set close to the site's latitude for maximum annual energy yield. Vastu Shastra specifies site orientations in degrees of compass bearing, with 0° = North.
Also known as
degrees to radiansradians to degreesangle conversiongrad to degreesdegree radian converterdegree to radian calculatorgradians converterarc minutes to degreesdeg to radangle unit converter