Angle Converter

What the Angle Converter does

This converter changes an angle between degrees, radians, gradians, turns, arc minutes, arc seconds, milliradians and degrees-minutes-seconds in one step. Enter 45 degrees and you see 0.785398 rad, which is π/4, plus 50 gon, 0.125 of a turn, 45° 0′ 0″ and the sine, cosine and tangent of the angle.

To convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π. The page does that for you and, when the angle is a whole-number fraction of 180°, also writes the radians as a multiple of π - the form maths teachers and textbooks expect. It also reads DMS angles from maps and GPS notes, such as 51° 28′ 38″ N.

How to use it

1. Type the angle and choose its unit. For degrees-minutes-seconds you can write 45 30 15, 45°30'15", 45:30:15 or add N, S, E or W; south and west become negative.
2. If you have radians as a multiple of π, choose Radians and tick the π box, then type 0.25 for π/4 or 1.5 for 3π/2.
3. Read the headline, then the table with every unit. Change the number of decimal places if you need more precision for a calculation or fewer for homework.
4. Below the table are the angle's type (acute, obtuse, reflex...), its quadrant, its reference angle, its complement and supplement, and sin, cos and tan - with exact surd values for 30°, 45°, 60° and their relatives.

Reading the results

Radians shown as a fraction of π, such as 3π/4, are exact; the decimal beside them is rounded. Use the π form in algebra and the decimal in a calculator or code.

Exact trigonometric values are given only where they exist in simple surds: multiples of 30° and 45°. Every other angle shows a rounded decimal. At 90° and 270° the tangent is undefined because the cosine is exactly zero, and the page says so rather than showing a huge number.

"Same as (0-360°)" is the direction the angle points after removing whole turns: 450° and -270° both point the same way as 90°. Trigonometric values only depend on that direction.

Complement and supplement are only shown when they are non-negative angles, which is how geometry courses define them.

Worked example: converting 135° for a trigonometry question

Formulas and scoring rules

Degrees to radians

radians = degrees × π / 180180° is exactly π radians, so the factor is exact; only the decimal display is rounded.

Radians to degrees

degrees = radians × 180 / π1 rad ≈ 57.295780°.

Degrees to gradians

gon = degrees × 400 / 360 = degrees × 10 / 9

Degrees to turns

turns = degrees / 360

DMS to decimal degrees

degrees = d + m / 60 + s / 3600Applied to the absolute value; the sign or hemisphere is applied afterwards.

Decimal degrees to DMS

d = floor(x); m = floor((x − d) × 60); s = (x − d − m/60) × 3600Seconds are rounded first and carried, so 59.999″ becomes the next minute rather than 60″.

Reference angle (0-360°)

Q1: θ; Q2: 180° − θ; Q3: θ − 180°; Q4: 360° − θ

Which unit is used where

Degrees divide a full turn into 360 and are what most people, maps and protractors use. Radians measure the arc length on a circle of radius 1, so a full turn is 2π; they are the SI unit and the one calculus and programming languages assume - Math.sin in JavaScript, Python and Excel's SIN all expect radians. Gradians (gon) divide a turn into 400, so a right angle is 100; some surveying instruments and calculators still offer them.

Arc minutes and arc seconds are sixtieths: navigation, astronomy and optics quote them, and GPS coordinates are often written in degrees, minutes and seconds. Milliradians are thousandths of a radian, common in rangefinding; note that some military scopes use a rounded "mil" of 1/6,400 of a turn, which is not the same as the true milliradian shown here.

Limitations: what the result does not prove

• Decimal results are rounded to the places you choose; carry the π form or more decimals into further calculations to avoid rounding drift.
• Exact surd forms are only produced for multiples of 30° and 45°. Angles such as 15° or 18° also have exact forms, but they are shown as decimals here.
• The NATO and other military mils (6,400 or 6,000 per turn) are not offered; the milliradian here is the exact 1/1,000 radian.
• The page converts angles; it does not convert map coordinates between datums or grid systems.

Privacy: where your data goes

Everything you paste, type or drop is processed in this browser tab. It is not uploaded, logged, stored or sent to analytics. Session recording and tag-manager scripts are switched off on this page.

Standards and sources

• NIST SP 811 - Guide for the Use of the International System of Units - checked 19 Sep 2026
• BIPM - The International System of Units (SI Brochure), radian definition
• NIST Digital Library of Mathematical Functions - 4.14 Definitions and periodicity of trigonometric functions

Frequently asked questions

Last reviewed 20 October 2026 by the A2Z.Tools team against the sources listed above.