Angle Converter
Convert degrees, radians, and gradians.
Understanding Angle Converter
An angle measures the amount of rotation between two lines, rays, directions, or positions around a common point. Angles can be expressed using several different units, with degrees, radians, and gradians being among the most common. Although these units use different numerical scales, they represent the same geometric quantity.
Degrees are widely used in everyday geometry, navigation, surveying, and many practical applications. Radians are the standard angular unit used throughout higher mathematics, calculus, physics, and many engineering calculations. Gradians, sometimes called gon or grads, divide a full rotation into 400 equal parts and are particularly associated with surveying and some technical applications.
How to get a useful estimate
Angle Converter is designed to make unit changes easier to verify without doing the conversion by hand. Start with the quantity and source unit, then select the destination unit when the calculator provides that choice. The available units or options include Degrees (°), Radians (rad), Gradians (grad), Degrees (°).
What can change the answer
A conversion changes the way a quantity is expressed; it does not change the underlying quantity. Pay attention to squared or cubed units, prefixes such as kilo- or milli-, and regional measurement conventions. Keeping the original value visible while checking the converted value is a simple way to catch a mistaken unit selection.
Convert angles between degrees, radians, and gradians. These units describe the same angular measurement using different numerical scales.
About Angle Converter
An angle measures the amount of rotation between two lines, rays, directions, or positions around a common point. Angles can be expressed using several different units, with degrees, radians, and gradians being among the most common. Although these units use different numerical scales, they represent the same geometric quantity.
Degrees are widely used in everyday geometry, navigation, surveying, and many practical applications. Radians are the standard angular unit used throughout higher mathematics, calculus, physics, and many engineering calculations. Gradians, sometimes called gon or grads, divide a full rotation into 400 equal parts and are particularly associated with surveying and some technical applications.
Angle Units
A complete rotation can be represented in three equivalent ways: 360 degrees, 2π radians, or 400 gradians. These relationships provide the basis for converting between the three units.
180° = π rad = 200 grad
90° = π/2 rad = 100 grad
How Angle Conversion Works
The calculator uses radians as the common reference unit. The original angle is first converted to radians and then converted from radians into the selected target unit. This approach allows degrees, radians, and gradians to be converted consistently.
rad = degrees × π / 180
Radians → Degrees:
degrees = rad × 180 / π
Gradians → Degrees:
degrees = gradians × 0.9
Degrees → Gradians:
gradians = degrees × 10 / 9
Degrees
A degree divides a full rotation into 360 equal parts. Degrees are commonly used for geometric measurements, directions, bearings, navigation, and everyday descriptions of angles. A right angle is 90 degrees, while a straight angle is 180 degrees.
Radians
A radian is defined using the geometry of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the circle's radius. A complete rotation contains 2π radians.
Radians are especially important in calculus because many trigonometric formulas and derivatives assume that angles are measured in radians. For example, the derivative of sin(x) is cos(x) when x is measured in radians.
Gradians
A gradian divides a full rotation into 400 equal parts. Therefore, a right angle is exactly 100 gradians and a straight angle is 200 gradians. Gradians can be useful in surveying, land measurement, and certain engineering applications.
Important Angle Equivalents
1 rad = 180/π° ≈ 57.2958°
1 grad = 0.9°
1° = 10/9 grad ≈ 1.11111 grad
Angle Converter Example
Suppose you want to convert 180 degrees into radians. Using the degree-to-radian relationship:
Therefore, 180° = π radians, which is approximately 3.141593 radians.
The same angle can also be expressed as 200 gradians. The geometric angle is identical; only its numerical representation changes.
Angles in a Circle
Several standard angles are frequently used when working with circles and geometry. A quarter rotation is a right angle, a half rotation is a straight angle, and a complete rotation returns to the original direction.
Half Rotation: 180° = π rad = 200 grad
Three-Quarter Rotation: 270° = 3π/2 rad = 300 grad
Full Rotation: 360° = 2π rad = 400 grad
Frequently Asked Questions
How many degrees are in a full circle?
A full rotation contains 360 degrees.
How many radians are in a full circle?
A full rotation contains 2π radians, approximately 6.283185 radians.
How many gradians are in a full circle?
A full rotation contains 400 gradians.
What is 180 degrees in radians?
180 degrees is equal to π radians, approximately 3.141593 radians.
What is 90 degrees in radians?
90 degrees is equal to π/2 radians, approximately 1.570796 radians.
What is the difference between degrees and radians?
They are two different units for measuring the same quantity. Degrees divide a full rotation into 360 parts, while radians are based on the relationship between the arc length and the radius of a circle.
Why are radians important in mathematics?
Radians provide a natural relationship between angles and the geometry of a circle. They are therefore used extensively in calculus, trigonometry, physics, and engineering.
Are gradians the same as degrees?
No. They represent the same type of measurement but use different scales. A right angle is 90 degrees or 100 gradians.
Disclaimer
This calculator is intended for informational purposes only. While we strive for accuracy, we cannot guarantee that every result is correct for every situation.