Angle Converter — Degrees, Radians, Milliradians & More

Angle Measurement Systems

There are 7 major systems for measuring angles, each used in different fields:

Unit	Symbol	Full Circle	Right Angle	Primary Users
Degree	°	360	90	Everyone — everyday, construction, navigation
Radian	rad	2π (6.2832)	π/2 (1.5708)	Math, physics, programming (all trig functions)
Milliradian	mrad	6,283.2	1,570.8	Military snipers, PRS shooting, mil-dot scopes
NATO Mil	mil	6,400	1,600	US Army artillery, NATO forces
Minute of Arc	MOA / ′	21,600	5,400	Hunting scopes, astronomy, benchrest shooting
Gradian	gon / grad	400	100	Surveying, European civil engineering
Turn	rev / τ	1	0.25	Mechanical engineering, rotational physics

Conversion Formulas — Quick Reference

From → To	Formula	Example
Degrees → Radians	rad = deg × π/180	90° = 1.5708 rad
Radians → Degrees	deg = rad × 180/π	π rad = 180°
Degrees → Milliradians	mrad = deg × 17.4533	1° = 17.45 mrad
Milliradians → Degrees	deg = mrad ÷ 17.4533	1 mrad = 0.0573°
Degrees → NATO Mils	mil = deg × 17.778	1° = 17.78 mil
Degrees → MOA	MOA = deg × 60	1° = 60 MOA
Degrees → Gradians	grad = deg × 10/9	90° = 100 grad
Degrees → Turns	turns = deg ÷ 360	360° = 1 turn

Key Angle Values — All Units

Degrees	Radians	mrad	MOA	Gradians	sin	cos
0°	0	0	0	0	0	1
30°	π/6 (0.5236)	523.6	1,800	33.33	0.500	0.866
45°	π/4 (0.7854)	785.4	2,700	50.00	0.707	0.707
60°	π/3 (1.0472)	1,047.2	3,600	66.67	0.866	0.500
90°	π/2 (1.5708)	1,570.8	5,400	100.00	1	0
120°	2π/3 (2.0944)	2,094.4	7,200	133.33	0.866	−0.500
180°	π (3.1416)	3,141.6	10,800	200.00	0	−1
270°	3π/2 (4.7124)	4,712.4	16,200	300.00	−1	0
360°	2π (6.2832)	6,283.2	21,600	400.00	0	1

Who Uses Each Angle Unit?

Audience	Primary Unit	Why
Students (US)	Degrees → Radians	AP Calculus, precalculus, trigonometry courses
Programmers	Radians	All trig functions (Math.sin, cos, atan2) use radians
Military snipers	Milliradians	1 mrad = 3.6" at 100yd — easy range estimation
US Army artillery	NATO mils	6,400 per circle — divisible by powers of 2
Hunters (US)	MOA	1 MOA ≈ 1" at 100yd — intuitive for zeroing
Surveyors	Gradians	Right angle = 100 grad — metric-friendly
Game developers	Radians → Degrees	Engine uses radians, UI displays degrees
Navigators	Degrees (+ DMS)	Compass bearings: 0°–360°

What Is an Angle?

An angle is the measure of rotation between two lines (or rays) that share a common endpoint (called the vertex). Angles are fundamental to geometry, trigonometry, physics, engineering, and navigation. They describe everything from the tilt of a roof to the trajectory of a bullet to the rotation of a 3D game character.

The most common way to measure angles is in degrees, where a full rotation = 360°. The Babylonians chose 360 because it has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360), making division easy.

In mathematics and programming, radians are preferred because they simplify calculus — the derivative of sin(x) is cos(x) only when x is in radians. In military applications, milliradians are preferred because 1 mrad subtends exactly 1 unit per 1,000 units of distance.

How to Measure Angles

Common tools for measuring angles in the US:

• Protractor — measures 0°–180° (semicircular) or 0°–360° (full circle). Standard school tool.
• Speed square — triangle-shaped tool used by US carpenters for marking 0°–90° angles.
• Digital angle finder — electronic tool showing angles to 0.1° precision. Used in construction.
• Mil-dot scope reticle — measures angles in milliradians for range estimation. Used in shooting.
• Smartphone apps — accelerometer-based tools for measuring angles and incline.