Angle Converter
Convert any angle between degrees, radians, gradians, turns, and degrees-minutes-seconds. See every unit at once, including radians as a multiple of pi.
📐 What is an Angle Converter?
An angle converter changes an angle from one unit of measure to another. Angles can be expressed in several units: degrees, radians, gradians, and turns, plus the degrees-minutes-seconds notation used in navigation and astronomy. Each unit describes the same rotation, just with a different scale. A right angle, for instance, is 90 degrees, π/2 radians, 100 gradians, or one quarter of a turn.
Converting between angle units is a routine task in many fields. Students moving between geometry and calculus must switch between degrees and radians constantly, because trigonometric functions in calculus assume radians. Surveyors work in gradians, where a right angle is a tidy 100 units. Navigators and astronomers record positions in degrees-minutes-seconds. Engineers describe rotation in turns or revolutions per minute. A converter removes the arithmetic and the risk of error.
A common misconception is that a calculator set to the wrong angle mode gives small errors. In fact it gives completely wrong answers: sin(90) is 1 when 90 is in degrees but about 0.894 when 90 is treated as radians. Another misconception is that radians are only for advanced maths. They appear whenever arc length, angular velocity, or oscillations are involved, because the formulas are simplest in radians.
This converter takes a value in degrees, radians, gradians, or turns and instantly shows the equivalent in every other unit, including radians written as a multiple of π and the full degrees-minutes-seconds breakdown. The working is shown so you can see exactly how each conversion factor is applied.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Right Angle in Degrees
Convert 90 degrees to every unit
Example 2 — One Radian to Degrees
Convert 1 radian to every unit
Example 3 — Gradians to a Straight Angle
Convert 200 gradians to every unit
❓ Frequently Asked Questions
🔗 Related Calculators
How do you convert degrees to radians?
Multiply the angle in degrees by π/180. For example, 90 degrees × π/180 = 1.5708 radians, which is exactly π/2. This works because a full circle is 360 degrees and also 2π radians, so one degree equals π/180 radians. To go back, multiply radians by 180/π.
How do you convert radians to degrees?
Multiply the angle in radians by 180/π. For example, 1 radian × 180/π = 57.2958 degrees. Since 2π radians equals 360 degrees, one radian is 360/(2π) = 57.2958 degrees. This calculator shows the degree value along with gradians, turns, and degrees-minutes-seconds.
What is a gradian?
A gradian, also called a gon or grade, is an angle unit where a full circle is 400 gradians and a right angle is exactly 100 gradians. It was designed to fit the decimal system and is mainly used in surveying and some engineering fields. To convert degrees to gradians, divide by 0.9, since 1 gradian equals 0.9 degrees.
What is a turn or revolution?
A turn, also called a revolution or full rotation, is one complete circle, equal to 360 degrees, 2π radians, or 400 gradians. Fractions of a turn are common in engineering: a quarter turn is 90 degrees and a half turn is 180 degrees. To convert degrees to turns, divide by 360.
What is degrees-minutes-seconds (DMS)?
Degrees-minutes-seconds is a way of writing angles where each degree is split into 60 arcminutes (′) and each arcminute into 60 arcseconds (″). For example, 57.2958 degrees is 57° 17′ 44.81″. DMS is widely used in navigation, astronomy, and geographic coordinates, similar to how time splits hours into minutes and seconds.
Why are radians used instead of degrees?
Radians are the natural unit of angle in mathematics and physics because many formulas are simpler in radians. Arc length equals radius times angle, and the derivative of sine equals cosine, only when the angle is measured in radians. Degrees are more intuitive for everyday use, but radians make calculus and wave equations cleaner.
How many radians are in a full circle?
A full circle contains 2π radians, which is about 6.2832 radians. This comes from the definition of a radian as the angle that subtends an arc equal to the radius: the full circumference is 2π times the radius, so it spans 2π radians. Half a circle is π radians and a right angle is π/2 radians.
What is 1 radian in degrees?
One radian equals about 57.2958 degrees, or 57° 17′ 45″ in degrees-minutes-seconds. It is the angle at which the arc length equals the radius of the circle. Because it is not a whole number of degrees, angles are usually written as multiples of π when working in radians, such as π/2 for 90 degrees.
How do you express radians as a multiple of pi?
Divide the radian value by π. For example, 1.5708 radians ÷ π = 0.5, so the angle is 0.5π, or π/2. Common angles are neat multiples of π: 180 degrees is π, 90 degrees is 0.5π, and 60 degrees is one-third π. This calculator shows both the decimal radian value and the multiple of π.
What is the difference between an arcminute and an arcsecond?
An arcminute is one sixtieth of a degree, and an arcsecond is one sixtieth of an arcminute, so there are 3,600 arcseconds in a degree. They provide fine resolution for small angles in astronomy and navigation. For instance, the apparent size of a coin at a distance might be measured in arcminutes or arcseconds.