Angle Converter

Convert between degrees, radians, gradians, arcminutes, arcseconds, and full turns — for math, navigation, surveying, and engineering.

Angle

degrees, radians, gradians

From

Result

0.017453

1 ° = 0.01745329 rad

Popular conversions

What Is an Angle Converter?

An angle converter translates a measure of rotation or inclination between units — degrees to radians, gradians to degrees, arcminutes to arcseconds. An angle describes how far something has turned, and there are several systems: 360 degrees, 2π radians, or 400 gradians make a full circle. Each suits a different field, so converting between them is routine in math, engineering, and navigation.

This converter routes through the radian, the SI unit of angle and the one trigonometry and calculus require. It also handles the fine subdivisions — arcminutes (1/60 of a degree) and arcseconds (1/3600 of a degree) — used in astronomy, surveying, and GPS coordinates.

This is one category of the full Unit Converter — pair it with our percentage calculator or scientific calculator for related everyday maths.

How Angle Conversion Works

Everything routes through the radian

Each unit has a fixed radian factor. The converter turns your input into radians, then projects it into degrees, gradians, and the rest.

180 degrees = π radians

A half-circle is 180° or π radians, so degrees × π/180 = radians. This is the core math-and-engineering conversion.

Gradians split the circle into 400

A gradian (gon) makes a right angle exactly 100 gradians, convenient for surveying. A full circle is 400 gradians.

Arcminutes and arcseconds

One degree = 60 arcminutes; one arcminute = 60 arcseconds. Astronomy and GPS use these fine subdivisions.

Core Angle Conversion Factors

Multiply to reach radians; divide to come back. The degree-radian bridge uses π.

Degrees → radians

× π/180

Multiply degrees by π/180 (≈0.01745) to get radians. 180° = π rad.

Radians → degrees

× 180/π

Multiply radians by 180/π (≈57.2958) to get degrees. 1 rad ≈ 57.3°.

Degrees → gradians

× 10/9

Multiply degrees by 10/9 to get gradians. 90° = 100 gradians.

How to Use the Angle Converter

1
Enter the angle value
Type the angle you want to convert — a rotation, a slope, a bearing, a coordinate component.
2
Choose the 'from' unit
Pick degrees, radians, gradians, arcminutes, arcseconds, or turns.
3
Choose the 'to' unit
Select the target unit, or swap the two to reverse direction.
4
Read every unit at once
The all-units table shows the angle in degrees, radians, and gradians together for an instant check.

Key Angle Concepts

Radian

The SI angle unit: the angle subtended by an arc equal to the radius. A full circle is 2π radians; trig and calculus assume radians.

Degree

The everyday angle unit: 1/360 of a circle. Inherited from Babylonian astronomy, it's used for navigation, geometry, and everyday rotation.

Gradian

An angle unit dividing the circle into 400 parts, making a right angle 100 gon. Favoured in surveying because slopes map neatly to percentages.

Arcminute / arcsecond

Subdivisions of a degree: 60 arcminutes per degree, 60 arcseconds per arcminute. Used for star positions, GPS, and fine angular measurement.

Real-World Angle Conversions

📐

Trigonometry

A 45° angle is π/4 ≈ 0.785 radians. Calculators and code default to radians, so degree inputs must be converted first.

🧭

Navigation bearings

A 90° bearing (due east) is π/2 radians or 100 gradians. Compass headings are in degrees, surveying often in gradians.

🌟

Astronomy

The Moon spans about 30 arcminutes (0.5°) in the sky. Star positions and telescope precision are quoted in arcseconds.

📍

GPS coordinates

A latitude of 40°26′46″ uses degrees, arcminutes, and arcseconds. Decimal-degree GPS converts these subdivisions.

⚙️

Rotational motion

A motor turning 2 revolutions sweeps 720° or 4π radians. Physics uses radians for angular velocity and acceleration.

📏

Surveying slopes

A 45° slope is 50 gradians, a clean half of a right angle. Surveyors favour gradians for this neat percentage mapping.

Best Practices for Angle Conversion

✓Use radians for trig and calculus. Math functions (sin, cos) and derivatives assume radians. Convert degrees with ×π/180 before computing, or results will be wrong.
✓Set your calculator's angle mode. A calculator in degree mode gives different answers than radian mode. Confirm the mode matches your input unit.
✓Mind arcminutes and arcseconds. Coordinates and astronomy use 60ths within a degree. Treating 30 arcminutes as 30 degrees overstates the angle 60-fold.
✓Use gradians for surveying slopes. Gradians make a right angle 100 units, so slopes map cleanly to percentages — handy in land survey and civil work.
✓Keep angles within one turn when needed. Bearings and headings often wrap at 360° (2π rad). Normalise large angles modulo a full turn when direction is what matters.

Common Angle Conversion Mistakes

Feeding degrees into trig functions

Most calculators and programming languages expect radians. Passing degrees directly to sin or cos gives wildly wrong results.

Confusing arcminutes with degrees

An arcminute is 1/60 of a degree. Reading 30′ as 30° overstates a coordinate or angle by a factor of 60.

Wrong calculator angle mode

A calculator left in radian mode when you meant degrees (or vice versa) silently produces incorrect trig values.

Forgetting the π factor

Degree-radian conversion uses π/180. Converting with a plain factor and no π gives an angle off by about 57×.

Why Angle Conversion Matters

Angles drive trigonometry, navigation, surveying, astronomy, and rotational mechanics — and the field decides the unit: degrees for everyday and navigation, radians for math and physics, gradians for surveying, arcminutes and arcseconds for fine precision. The classic failure is feeding degrees into a function that expects radians, which produces answers that are not just imprecise but nonsensical.

Because angles feed directly into positions, slopes, and trig results, a unit slip propagates into coordinates, courses, and calculations. A converter that routes through the radian and surfaces degrees, gradians, and arcseconds together lets a student, navigator, or surveyor work in the right unit for the task and verify a value at a glance.

Built for students, navigators, surveyors, astronomers, and engineers converting between degrees, radians, and gradians.

Linear unit factors follow the BIPM SI brochure, the NIST Guide to the SI, and ISO 80000. Currency rates load live from open.er-api.com; crypto prices from CoinGecko. See our methodology and editorial policy. Educational only — not certified for regulated trading, settlement, medical, or aerospace use.

Angle Converter FAQs

Multiply degrees by π/180 (about 0.017453). So 90° = 90 × π/180 = π/2 ≈ 1.5708 radians, and 180° = π radians. To go the other way, multiply radians by 180/π (about 57.2958). Radians are required for trigonometry and calculus.

One radian is about 57.2958 degrees (180/π). A full circle is 2π radians = 360°, and a half-circle is π radians = 180°. The radian is defined by the arc length equalling the radius, which is why it's the natural unit for circular motion.

A gradian (also called a gon or grade) divides a full circle into 400 parts, so a right angle is exactly 100 gradians and 90° = 100 gon. Surveyors favour gradians because slopes map cleanly onto percentages — a 50-gradian slope is a 45° incline.

They're fine subdivisions of a degree: one degree contains 60 arcminutes (′), and one arcminute contains 60 arcseconds (″). So 1° = 3,600 arcseconds. Astronomy, surveying, and GPS coordinates use them to express very small or precise angles.

Usually because the angle mode doesn't match the input. Trig functions assume radians by default in most programming and many calculators; if you enter degrees while the tool expects radians (or vice versa), the result is wrong. Convert to the expected unit, or set the calculator's mode correctly.

It uses exact definitions (180° = π radians, 400 gradians = 360°, 60 arcminutes = 1°, 60 arcseconds = 1 arcminute) and routes every conversion through the radian at full precision, so the result is exact to your input precision.