Angle Converter

About this Angle Converter

The Angle Converter translates angular measurements between four common systems. It is useful in geometry, trigonometry, programming, navigation and technical calculations.

Degrees are the most familiar angle unit in everyday use, but radians are the standard unit in most higher mathematics, physics and programming trigonometric functions, so converting between them correctly is essential whenever you move between an everyday angle value and code or a scientific formula.

For related tasks, try Length Converter, Area Converter or Weight Converter.

How the calculation works

Conversions use radians as the internal base measure. One full turn equals 2π radians or 360 degrees.

How to use it

1. Enter the angle value.
2. Choose the source and target angle units.
3. Convert to see the equivalent.

Practical example

180 degrees equals π radians, 200 gradians and one-half turn.

90 degrees converts to exactly pi/2 radians, approximately 1.5708, and to 100 gradians, since gradians divide a right angle into exactly 100 units, making gradians convenient for some surveying calculations that favor decimal subdivisions.

Things to double-check

A full circle is 360 degrees, 2-pi radians (approximately 6.2832), 400 gradians, or exactly 1 turn; because radians involve pi, a converted value will usually be a long, non-terminating decimal unless the original degree value happens to be a clean fraction of 360.

Degrees, radians, gradians and turns

A full circle is 360 degrees, 2 pi radians (about 6.2832), 400 gradians or one turn. One radian is about 57.2958 degrees, so 45 degrees is 0.7854 radians and 3 radians is about 171.89 degrees. A right angle is 90 degrees, pi over 2 radians or 100 gradians. Degrees are the everyday unit, radians are the natural unit in mathematics, gradians are used in some surveying, and turns are convenient when you think in whole rotations. The converter lets you move between them without memorising these factors.

Radians in code and calculus

Most programming languages, including JavaScript, expect angles in radians for sine, cosine and tangent. Passing 90 to a sine function expecting radians will give the wrong answer, so convert first: 90 degrees is pi over 2. Calculus formulas, such as the derivative of sine being cosine, only work cleanly when angles are in radians. If a result from a script looks odd, an angle unit mismatch is one of the first things to check.

Slopes, pitches and bearings

Angles describe slopes in several ways. A gradient of 100 percent is a 45 degree slope, and a 5 degree slope is a grade of about 8.75 percent, found from the tangent of the angle. Roof pitch is often given as rise over run, and road signs use percent grades. Navigation uses compass bearings measured in degrees clockwise from north, from 0 to 360. Converting units helps when plans, tools and software use different conventions.

Check your results

A simple sanity check is to remember a few anchors: 30 degrees is about 0.5236 radians, 60 degrees is about 1.0472, 180 degrees is 3.1416 and 360 degrees is 6.2832. If you convert a value and it is not close to what you expect from these anchors, recheck the units you selected. Very large angles beyond 360 degrees are valid and simply represent more than one rotation, which is useful for rotating machinery and animation.

Angles on a clock face

A clock is a handy way to feel angles. The minute hand moves 6 degrees every minute, and the hour hand moves half a degree each minute, or 30 degrees each hour. At 3:00 the hands are 90 degrees apart. At 2:30 the hour hand is at 75 degrees and the minute hand at 180 degrees, so the angle between them is 105 degrees. Converting that to radians gives about 1.83. Problems like this are common in school mathematics and aptitude tests, and the converter lets you check each step.

Slopes and ramps

Angles describe ramps and roofs. A roof pitch of 6 in 12 is an angle of about 26.6 degrees, and a wheelchair ramp with a gradient of 1 in 12 is about 4.8 degrees. One degree of latitude on the Earth's surface is about 111 kilometres. When a plan gives a slope as a ratio and a tool wants degrees, convert using the inverse tangent of rise over run. These examples show how angle units connect to everyday measurements.

Worked examples in four units

To convert 2.5 radians to degrees, multiply by 180 over pi, which is about 57.2958, to get about 143.24 degrees. To convert 215 degrees to radians, multiply by pi over 180, which gives about 3.752 radians. A turn of 0.35 equals 0.35 times 360, or 126 degrees, and 150 gradians equals 150 times 0.9, or 135 degrees. Keep a note of the two key factors: 180 degrees equals pi radians, and 100 gradians equals 90 degrees. With those two facts you can convert any angle by hand and check the calculator's result.

Radians, arcs and sectors

Radians are the natural unit when you work with circles because the arc length equals the radius times the angle in radians. A radius of 2 metres and an angle of 1.2 radians gives an arc of 2.4 metres, and the area of the sector is half the radius squared times the angle, which is 2.4 square metres. If you tried to use degrees in the same formulas, you would need an extra factor of pi over 180. This is why physics, engineering and programming formulas almost always use radians, and why converting degrees first is the safe habit.

Bearings, compasses and direction

Navigation uses angles measured clockwise from north, from 0 to 360 degrees. East is 90, south is 180 and west is 270. A back bearing, the direction to return along the same line, is found by adding or subtracting 180 degrees, so the back bearing of 70 is 250. Compass points such as north-east correspond to 45 degrees. Remember that magnetic north differs from true north by an amount that changes by place, so a compass bearing needs a correction when you plot it on a map. Converting between degrees and radians matters when you enter directions into mapping software.

Angular speed and mechanical uses

Rotating machines are described by angular speed. A motor turning at 3,000 revolutions per minute has an angular speed of 3,000 times 2 pi divided by 60, which is about 314.2 radians per second. Gears, wheels, antenna dishes and cameras on pan and tilt heads all depend on angle conversions. In photography, the field of view is an angle, and a longer lens narrows it. When you read a specification, check whether it uses degrees, radians or turns, and convert before you plug the value into a formula. A wrong unit can cause an error by a factor of about 57.

Key terms explained

A degree is one 360th of a full rotation. A radian is the angle at the centre of a circle that cuts off an arc equal in length to the radius, and a full circle is 2 pi radians. A gradian, or gon, divides a right angle into 100 parts. A turn is one complete rotation. An acute angle is less than 90 degrees, a right angle is exactly 90, an obtuse angle is between 90 and 180, and a straight angle is 180. Minutes and seconds of arc are sixtieths of a degree, used in navigation and astronomy.

Rules of thumb and quick checks

Remember that pi radians is 180 degrees, so one radian is about 57.3 degrees. To convert degrees to radians, multiply by 0.01745, and to convert radians to degrees, multiply by 57.2958. Handy anchors are 30 degrees as 0.524, 45 as 0.785, 60 as 1.047 and 90 as 1.571 radians. A turn of one quarter is 90 degrees and 100 gradians. If a trigonometric function in code gives an odd result, check whether the angle was supposed to be in radians.

Frequently asked questions