Angle Converter

What Is an Angle?

An angle measures the amount of rotation between two lines or planes that share a common point (called the vertex). Angles are fundamental to geometry, trigonometry, navigation, construction, physics, and engineering.

A full rotation around a point is 360 degrees (°), which is equivalent to 2π radians, 400 gradians, or 1 revolution. Our converter above supports all 9 standard angle units and shows the value in every unit simultaneously.

Angle Units Explained

Degrees (°)

The most common angle unit. A full circle is 360°. The number 360 was likely chosen by ancient Babylonian astronomers because it has 24 divisors, making it easy to subdivide. Degrees are the SI accepted unit for angle and are used in everyday life, education, construction, and navigation.

Degrees can be further divided into arcminutes (′) and arcseconds (″). There are 60 arcminutes in 1 degree and 60 arcseconds in 1 arcminute (so 3,600 arcseconds per degree). This notation — called DMS (degrees, minutes, seconds) — is used in GPS coordinates and astronomy.

Radians (rad)

The SI derived unit for angle. A radian is defined as the angle subtended by an arc whose length equals the radius of the circle. A full circle is 2π ≈ 6.2832 radians. Radians are essential in calculus, physics, and engineering because trigonometric functions (sin, cos, tan) work most naturally in radians.

Key values: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°, 2π = 360°.

Gradians (grad/gon)

Used primarily in surveying and land measurement. A full circle is 400 gradians, making right angles exactly 100 grad — convenient for survey calculations. Also called gons or grades. One gradian equals 0.9 degrees.

Milliradians (mrad)

One-thousandth of a radian. Used extensively in military ballistics, optics, and telescopic sights. At 1,000 meters, 1 milliradian subtends approximately 1 meter — making distance estimation intuitive. A full circle is approximately 6,283 milliradians.

Mils (NATO)

The NATO mil divides a circle into 6,400 mils. Used in military artillery and navigation for its practical divisibility. Not to be confused with milliradians — though similar in purpose, they differ slightly in value: 1 NATO mil = 0.05625° (vs. 1 mrad ≈ 0.05730°).

Revolutions, Circles, Arcminutes & Arcseconds

A revolution (or circle) is one complete rotation = 360°. Used in rotational mechanics (RPM = revolutions per minute). Arcminutes (1° = 60′) and arcseconds (1′ = 60″) provide precision for astronomy, GPS coordinates, and geodesy.

How to Convert Degrees to Radians

Since π radians = 180°, the conversion formula is:

radians = degrees × π / 180

Example: Convert 90° to radians.

• radians = 90 × π / 180
• radians = 90π / 180
• Simplify: 90/180 = 1/2
• radians = π/2 ≈ 1.5708 rad

How to Convert Radians to Degrees

The reverse formula is:

degrees = radians × 180 / π

Example: Convert 1 radian to degrees.

• degrees = 1 × 180 / π
• degrees = 180 / 3.14159…
• degrees ≈ 57.2958°

Common Angle Conversion Table

Degrees (°)	Radians (rad)	Gradians (grad)	Revolutions
0°	0	0	0
30°	π/6 ≈ 0.5236	33.333	1/12
45°	π/4 ≈ 0.7854	50	1/8
60°	π/3 ≈ 1.0472	66.667	1/6
90°	π/2 ≈ 1.5708	100	1/4
120°	2π/3 ≈ 2.0944	133.333	1/3
180°	π ≈ 3.1416	200	1/2
270°	3π/2 ≈ 4.7124	300	3/4
360°	2π ≈ 6.2832	400	1

Frequently Asked Questions

Why are radians used instead of degrees in math?

Radians simplify formulas in calculus and physics. The derivative of sin(x) is cos(x) only when x is in radians. In degrees, there would be an extra factor of π/180 in every derivative and integral. Radians also connect arc length directly to angle: arc length = radius × angle (in radians).

How many radians are in a full circle?

A full circle is 2π radians (approximately 6.2832 radians). This follows from the definition: the circumference of a circle is 2πr, and dividing by the radius r gives 2π.

What is a gradian and when is it used?

A gradian (also called a gon) divides a right angle into exactly 100 parts, making a full circle 400 gradians. This decimal-friendly system is used primarily in surveying, civil engineering, and in some European countries for land measurement. Many survey calculators and total stations support gradians natively.

What is the difference between a mil and a milliradian?

A milliradian is exactly 1/1000 of a radian (≈ 0.05730°). A NATO mil divides the circle into 6,400 parts (= 0.05625°). They are close but not identical. The NATO mil was chosen for military use because 6,400 is easily divisible by powers of 2, simplifying field calculations.

How do arcminutes and arcseconds relate to degrees?

1 degree = 60 arcminutes = 3,600 arcseconds. This sexagesimal (base-60) system was inherited from Babylonian mathematics. GPS coordinates use DMS notation: 40° 26′ 46″ N means 40 degrees, 26 arcminutes, 46 arcseconds north latitude.

Can I measure angles with a protractor?

Yes — a protractor is the standard tool for measuring angles in degrees. Place the center point on the vertex of the angle, align one side with the base line (0°), and read where the other side crosses the scale. Protractors typically measure 0°–180° (half-circle) or 0°–360° (full-circle).