Angle Converter

Frequently Asked Questions

How do degrees and radians relate?

180° = π radians, so radians = degrees × π/180 and degrees = radians × 180/π. A right angle is 90° = π/2 rad ≈ 1.5708 rad.

What is a gradian?

Also called a gon: a right angle is 100 gradians, so a full circle is 400 gon. It is used in surveying and some European engineering. 1 gon = 0.9°.

What are arcminutes and arcseconds?

Subdivisions of a degree: 1° = 60 arcminutes (′), 1′ = 60 arcseconds (″). They are used in astronomy, navigation, and optics for very small angles.

When does the π-form appear?

For clean degree inputs the tool also shows the exact radian value as a fraction of π (e.g., 90° → π/2, 270° → 3π/4), which is the form used in math and physics.

Provided by AllCalculators.io
Free online calculators for everyday. No registration required.

Estimates for informational purposes only.

Important Disclaimer: Estimates for informational purposes only.

This calculator provides estimates for informational purposes only. Results are based on assumptions and may not reflect actual outcomes. Consult qualified professionals in relevant fields before making important decisions based on these results.

JavaScript is required to use the interactive calculator above. The questions and answers below remain readable without JavaScript.

How This Calculator Works

An angle measures rotation, and one full turn can be named many ways: 360 degrees, 2π radians, 400 gradians, or 1 revolution. Each unit is defined as a fixed number of degrees, so conversion pivots through the degree: result = amount × (degreesPerFrom ÷ degreesPerTo). The factors used are: 1 radian = 180/π ≈ 57.29578°, 1 gradian (gon) = 0.9°, 1 arcminute = 1/60°, 1 arcsecond = 1/3600°, 1 revolution = 360°, and 1 milliradian = (180/π)/1000 °. When the value resolves to a clean fraction of 180°, the converter also returns the exact symbolic radian form - for example 90° = π/2 rad and 270° = 3π/2 rad - which is what trigonometry and calculus actually want rather than a rounded decimal.

Worked Example

Convert 90° to radians. 90 × (π/180) = π/2 ≈ 1.5707963 rad, and the exact form π/2 is shown. Convert 1 revolution: 1 × 360 = 360° = 2π rad = 400 gon. Convert 1.5 mrad (a common rifle-scope click) to arcminutes: 1.5 mrad × (180/π)/1000 ≈ 0.085944°, then ÷ (1/60) ≈ 5.156 arcminutes - close to the 5.4 MOA rule-of-thumb gunners use. Convert 30 arcseconds to degrees: 30 × (1/3600) = 0.008333°.

Where You Use This

Astronomers express tiny separations in arcminutes and arcseconds (the Moon spans about 30 arcminutes). Surveyors and European engineers use gradians, where a right angle is a tidy 100 gon. Programmers must feed radians to Math.sin and Math.cos while their UI shows degrees. Riflery and ballistics use milliradians and MOA for scope adjustments. CAD, robotics, and CNC work mix degrees and radians constantly, and navigation uses revolutions and degrees for headings.

Common Mistakes

Feeding degrees to trig functions: nearly every math library expects radians; sin(90) is not 1.

Gradian ≠ degree: a right angle is 90° but 100 gon - mixing them throws surveys off by 11%.

Arcminute vs minute of time: 1 arcminute = 1/60 of a degree, unrelated to clock minutes despite the name.

Dropping the π: 180° is π rad ≈ 3.14159, not 180; forgetting the π factor is the classic conversion bug.

Milli- prefix: 1 mrad = 0.001 rad ≈ 0.0573°; treating mrad as rad inflates angles 1000×.

Reference Table

Full turn = 360° = 2π rad = 400 gon = 1 rev = 6283.19 mrad

Straight angle = 180° = π rad = 200 gon

Right angle = 90° = π/2 rad = 100 gon

1 rad = 57.29578° = 63.66198 gon = 3437.75 arcmin

1° = 60 arcmin = 3,600 arcsec = 17.4533 mrad

1 gon = 0.9° = 0.015708 rad

30° = π/6, 45° = π/4, 60° = π/3, 270° = 3π/2

Frequently Asked Questions

Related Conversions Calculators