Angle Converter

What is Angle and Why Do We Convert It?

An angle measures the amount of rotation between two lines or surfaces meeting at a point. Angles are fundamental in geometry, trigonometry, navigation, architecture, astronomy, and engineering. Different fields use different units to express angles: degrees are used in everyday geometry and navigation, radians are used in mathematics and physics, gradians are used in surveying and civil engineering, and minutes/seconds of arc are used in astronomy and precise geographic measurements. Converting between these is essential for accurate calculations.

Angle Unit Conversion Formulas

Quick reference for the most commonly used angle conversions:

full circle = 360 degrees (°) = 2π radians = 400 gradians = 1 turn

degree (°) = π/180 radians ≈ 0.017453 radians

radian = 180/π degrees ≈ 57.2958°

degree (°) = 60 minutes of arc (′)

minute of arc (′) = 60 seconds of arc (″)

degree (°) = 3,600 seconds of arc (″)

gradian (grad) = 0.9 degrees = π/200 radians

turn = 360° = 2π radians = 400 gradians

Common Angle Conversions — Reference Chart

From	To	Multiply By
Degree (°)	Radian	0.017453 (π/180)
Radian	Degree (°)	57.2958 (180/π)
Degree (°)	Gradian	1.11111
Gradian	Degree (°)	0.9
Degree (°)	Minute of Arc (′)	60
Minute of Arc (′)	Second of Arc (″)	60
Radian	Gradian	63.6620 (200/π)
Turn	Degree (°)	360

Common Angles — Reference Values

Angle Name	Degrees (°)	Radians	Gradians
Zero angle	0°	0	0 grad
Right angle	90°	π/2 ≈ 1.5708	100 grad
Straight angle	180°	π ≈ 3.14159	200 grad
Full circle	360°	2π ≈ 6.28318	400 grad
30° (trigonometry)	30°	π/6 ≈ 0.5236	33.33 grad
45° (diagonal)	45°	π/4 ≈ 0.7854	50 grad
60° (equilateral triangle)	60°	π/3 ≈ 1.0472	66.67 grad

Real World Examples of Angle Conversion

Navigation: A compass bearing of 90° (due East) = π/2 radians = 100 gradians
Architecture: A roof slope of 30° = 0.5236 radians — common in Indian house design
Astronomy: The Moon's apparent diameter ≈ 0.5° = 30 arcminutes = 1,800 arcseconds
Surveying: Civil engineers in Europe use gradians — a right angle = exactly 100 gradians
Trigonometry: sin(π/6) = sin(30°) = 0.5 — radians essential for calculus formulas
GPS precision: 1 arcsecond of latitude ≈ 31 meters on Earth's surface
Clock hands: Minute hand moves 6° per minute = 0.1047 radians per minute

Angle Units Explained

Unit	Symbol	Full Circle	Used In
Degree	°	360°	Everyday geometry, navigation, maps
Radian	rad	2π rad	Mathematics, physics, calculus
Gradian	grad or gon	400 grad	Surveying, civil engineering (Europe)
Minute of Arc	′ (arcmin)	21,600′	Astronomy, navigation, GPS
Second of Arc	″ (arcsec)	1,296,000″	Astronomy, telescopes, precision maps
Turn	tr or rev	1 turn	Rotational motion, engineering

Frequently Asked Questions

How do I convert degrees to radians?

Multiply by π/180 ≈ 0.017453. For example, 90° = 90 × π/180 = π/2 ≈ 1.5708 radians. Radians are the natural unit for trigonometric functions in mathematics and physics.

How do I convert radians to degrees?

Multiply by 180/π ≈ 57.2958. For example, 1 radian = 57.2958°. π radians = 180°, and 2π radians = 360° (full circle).

What is a gradian (grad)?

A gradian divides a full circle into 400 equal parts, so a right angle = exactly 100 gradians. This makes calculations cleaner in surveying. Gradians are also called "gons" and are the standard in many European surveying instruments.

What is an arcminute and arcsecond?

1 degree = 60 arcminutes (′) and 1 arcminute = 60 arcseconds (″). These tiny subdivisions are used in astronomy (measuring star positions) and GPS coordinates. For example, your GPS location might be expressed as 28°36′17″N.

Why do mathematicians prefer radians over degrees?

Radians make calculus formulas simpler. For example, the derivative of sin(x) = cos(x) only works when x is in radians. In degrees, extra conversion factors would be needed throughout every formula.

How many degrees in one radian?

1 radian = 180/π ≈ 57.2958 degrees. This is why π radians = 180° and 2π radians = 360°.

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