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Angle Converter

Measurement

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

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.

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
What is an angle converter?
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.
How do I convert degrees to radians?
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.
What is a radian?
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.
What is a gradian and who uses it?
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.
What are arcminutes and arcseconds?
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.
What is a milliradian (mrad) and where is it used?
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.
How do I convert GPS coordinates from decimal degrees to degrees-minutes-seconds?
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.
How many degrees is a full circle, and what are common reference angles?
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).
How do I use the Angle Converter?
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.
Why do programming languages use radians instead of degrees?
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.
What is the angular resolution of the human eye?
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.
What angles are commonly used in Indian architecture and construction?
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.

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