Conversion Runs locally Ready to use Built-in examples No tracking

Angle Converter

Convert between degrees, radians, gradians (400 per turn) and turns — 180° = π rad ≈ 3.14159, a right angle = 100 gon and one full turn = 360° = 2π rad = 400 gon. Handy for maths homework, land surveying and CAD work.

Maths problems mix degrees and radians, surveyors work in gradians with 400 to the full circle, and game engines and animation work in turns — four units share the same angle. This tool converts between degrees (°), radians (rad), gradians (gon) and turns, giving all four rows for a single input; the relationships underneath are 180° = π rad = 200 gon = 0.5 turn.

The easy mistakes: the radian is dimensionless, being a ratio of arc length to radius, and it is the default in mathematical derivation; the gradian is the European surveying habit, with a right angle at 100 gon, which is convenient in decimal; and a calculator left in DEG rather than RAD mode will be out by a factor of more than 57 when evaluating a trigonometric function, so check which one a function expects before passing a value in.

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How to use

  1. Enter the angle in any one of the four units.
  2. Read the other three alongside it.
  3. Confirm whether the receiving function expects degrees or radians.
  4. Use 180° = π rad = 200 gon = 0.5 turn as a sanity check.

How it works

Basic usage

Enter a value, pick the source and target units, and it converts live; the swap button reverses direction in one click.

Degrees and radians

The radian is the mathematical standard: 180 deg = pi rad, 90 deg = pi/2 rad, 45 deg = pi/4 rad.

Gradians (gon)

The gradian (gon) uses a 400-division circle: a right angle = 100 gon, used in some European engineering surveying; turns suit rotation and spin scenarios (0.25 turn = 90 deg).

Code example

JavaScript Four units, with degrees as the hub 

const toDeg = { deg: 1, rad: 180 / Math.PI, gon: 0.9, turn: 360 };
const fromDeg = { deg: 1, rad: Math.PI / 180, gon: 10 / 9, turn: 1 / 360 };

function convert(v, from, to) {
  return v * toDeg[from] * fromDeg[to];
}

convert(180, "deg", "rad");   // 3.141592653589793
convert(0.25, "turn", "deg"); // 90

Python Built-in conversions in the math module 

import math

math.degrees(math.pi)   # 180.0 (radians to degrees)
math.radians(180)       # 3.14159... (degrees to radians)

# Gradians and turns are a manual multiplier:
180 * 10 / 9    # 200 gon
0.5 * 360       # 180 degrees

FAQ

How do radians and degrees convert?

180 deg = pi rad, so 1 deg = pi/180, about 0.0174533 rad, and 1 rad = 180/pi, about 57.2958 deg. Multiply degrees by pi/180 to get radians.

What is the gradian (gon)?

The gradian (gon, also grad) divides the circle into 400 equal parts, so a right angle = 100 gon — convenient for decimal calculation. European geodesy and some engineering fields still use it.

Why do sin results look wrong in Excel or a calculator?

Most likely the degree/radian mode is reversed. Math functions default to radians, so the sin of 30 deg requires converting first to 0.5236 rad. Convert here, then compute.

How do degrees and radians convert exactly?

A full turn of 360 deg corresponds to 2pi radians, so 180 deg = pi rad, about 3.14159. Formula: radians = degrees x pi / 180, degrees = radians x 180 / pi. Key landmarks: 30 deg = pi/6, 45 = pi/4, 60 = pi/3, 90 = pi/2.

Why use radians instead of degrees?

Because calculus and physics formulas hold only in radians: the limit of sin x / x = 1, arc length l = r*theta, and angular velocity omega = dtheta/dt all assume radians. Degrees are an arbitrary division; the radian is the ratio of arc length to radius, a dimensionless pure number.

Where are gradians used?

A gradian divides the circle into 400 parts, so a right angle is exactly 100 gon and a straight angle 200 gon. Right and straight angles being integers eases surveying and land-measurement arithmetic, so European countries like France and Switzerland still use it in surveying.

How do angular and linear velocity convert?

Linear velocity v = angular velocity omega x radius r, provided omega is in radians per second. For example a wheel of radius 0.3 m spinning at 10 rad/s has a linear velocity of 3 m/s. This is why precise conversion favors radians — the formula needs no extra coefficient.