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Angle Converter: Convert Degrees, Radians, Gradians, Minutes, Seconds and More

Converting angles between degrees, radians, gradians, minutes, seconds, and revolutions is common in mathematics, geometry, engineering, navigation, surveying, and physics. The calculations are usually simple, but remembering the right relationship between each unit can become frustrating when you are working with several different measurements.

The CalculatorKits Angle Converter lets you convert angle values instantly. Enter your number, choose the original unit, select the unit you need, and the converted result appears immediately.

The tool supports more than 15 angle units and uses radians as its internal base unit. It also provides common presets, live results, and a copy option for convenient use.

Quick Answer

These are some of the most useful angle conversions:

Conversion	Result
1 radian	57.2958°
1 degree	0.0174533 rad
90°	π/2 rad ≈ 1.5708 rad
180°	π rad ≈ 3.14159 rad
360°	2π rad ≈ 6.28319 rad
1 gradian	0.9°
1 revolution	360°
1 degree	60 arcminutes
1 arcminute	60 arcseconds
1°	0.0174533 rad
45°	π/4 rad ≈ 0.785398 rad

NIST lists the standard conversion factors for degrees, radians, gradians, minutes, seconds, revolutions, and other angle units.

What Is an Angle Converter?

An Angle Converter is a tool that changes an angle from one measurement unit into another without requiring you to calculate the conversion manually.

For example, you may have an angle of 180 degrees but need the answer in radians for a trigonometry or calculus formula.

The conversion is:

180° = π radians

Or approximately:

180° = 3.14159 rad

The physical or geometric angle has not changed. Only the way it is expressed has changed.

Common angle units include:

1. Degrees
2. Radians
3. Gradians or gons
4. Arcminutes
5. Arcseconds
6. Revolutions or turns
7. Quadrants
8. Milliradians
9. Other specialized and historical angle units

The CalculatorKits converter supports more than 15 angle units, allowing the same tool to handle everyday, mathematical, engineering, and technical conversions.

Use the CalculatorKits Angle Converter

The easiest way to convert an angle is to use the CalculatorKits Angle Converter.

The tool is designed around a simple From and To workflow.

How to Convert an Angle

1. Enter the angle value.
2. Select the unit you are converting from.
3. Select the unit you want as the result.
4. View the converted value.
5. Copy the result if you need to use it elsewhere.

For example, to convert 180 degrees to radians, enter 180, choose degrees as the starting unit, and radians as the target unit.

The result is approximately:

180° = 3.14159 rad

The tool performs its calculations in the browser and uses radians as the base unit for its conversion system.

What Is a Radian?

A radian is the coherent SI unit for plane angle.

It is defined using the relationship between the length of an arc and the radius of a circle. If the arc length is equal to the radius, the angle formed at the center is one radian.

The relationship can be written as:

θ = s ÷ r

where:

θ = angle in radians

s = arc length

r = radius

One complete circle contains:

2π radians

Therefore:

2π rad = 360°

The BIPM and NIST identify the radian as the coherent SI unit for plane angle.

Degrees to Radians

Degrees are probably the most familiar way of measuring angles in everyday geometry.

To convert degrees to radians, use:

Radians = Degrees × π ÷ 180

For example:

90° × π ÷ 180 = π/2

Therefore:

90° = 1.5708 rad

Common Degrees to Radians

Degrees	Radians
0°	0
30°	0.523599
45°	0.785398
60°	1.0472
90°	1.5708
120°	2.0944
135°	2.35619
180°	3.14159
270°	4.71239
360°	6.28319

The exact relationship comes from:

360° = 2π rad

NIST gives the degree to radian factor as approximately 0.01745329 rad per degree.

Radians to Degrees

To convert radians to degrees, use:

Degrees = Radians × 180 ÷ π

For example:

1 rad × 180 ÷ π ≈ 57.2958°

Therefore:

1 radian ≈ 57.2958 degrees

Common Radian to Degree Conversions

Radians	Degrees
0	0°
π/6	30°
π/4	45°
π/3	60°
π/2	90°
2π/3	120°
3π/4	135°
π	180°
3π/2	270°
2π	360°

For exact mathematics, it is often better to retain π rather than immediately converting it to a decimal.

For example:

π/4 rad = 45°

is more precise than writing only 0.785398 rad.

180 Degrees to Radians

This is one of the most frequently searched angle conversions.

Using:

Radians = Degrees × π ÷ 180

we get:

180 × π ÷ 180 = π

Therefore:

180° = π radians

Approximately:

180° = 3.14159 radians

This relationship is especially important in trigonometry and calculus.

90 Degrees to Radians

A right angle is 90 degrees.

Convert it using:

90 × π ÷ 180 = π/2

Therefore:

90° = π/2 radians

Approximately:

90° = 1.5708 radians

45 Degrees to Radians

A 45 degree angle is:

45 × π ÷ 180 = π/4

Therefore:

45° = π/4 radians

Approximately:

45° = 0.785398 radians

Gradians to Degrees

Gradians are also called gons or grades in some technical references.

A complete circle contains:

400 gradians

while the same circle contains:

360 degrees

Therefore:

1 gradian = 0.9°

The conversion formula is:

Degrees = Gradians × 0.9

For example:

100 gradians × 0.9 = 90°

So:

100 grad = 90°

NIST gives the same 0.9 degree relationship for a gon.

Common Gradian Conversions

Gradians	Degrees
50	45°
100	90°
150	135°
200	180°
250	225°
300	270°
400	360°

Gradians can be particularly useful in surveying and some engineering applications because a right angle is exactly 100 gradians.

Degrees to Gradians

The reverse relationship is:

Gradians = Degrees × 10 ÷ 9

For example:

90° × 10 ÷ 9 = 100 gradians

Therefore:

90° = 100 grad

Another example:

45° × 10 ÷ 9 = 50 grad

So:

45° = 50 grad

Degrees, Minutes and Seconds

Angles are not always written as decimal degrees.

In navigation, surveying, geography, astronomy, and related applications, an angle may be written using degrees, minutes, and seconds.

This system is commonly called DMS, meaning:

Degrees, Minutes, Seconds

The relationships are:

1 degree = 60 minutes

1 minute = 60 seconds

Therefore:

1 degree = 3,600 seconds

The symbols are:

Degrees = °

Minutes = ′

Seconds = ″

For example:

25° 30′ 00″

means 25 degrees and 30 minutes.

Because 30 minutes is half of a degree:

25° 30′ = 25.5°

Decimal Degrees to DMS

Suppose you have:

25.75°

The whole number is:

25°

The remaining:

0.75° × 60 = 45′

So:

25.75° = 25° 45′ 00″

Another example is:

42.125°

The fractional part is:

0.125 × 60 = 7.5 minutes

That gives:

42° 7′ 30″

This is useful when converting coordinates or technical angle measurements between decimal and DMS formats.

DMS to Decimal Degrees

To convert degrees, minutes, and seconds into decimal degrees:

Decimal Degrees = Degrees + Minutes ÷ 60 + Seconds ÷ 3600

For example:

30° 15′ 30″

becomes:

30 + 15/60 + 30/3600

= 30 + 0.25 + 0.008333

= 30.258333°

The calculation is straightforward, but mistakes are easy when several DMS values must be converted. An angle converter can reduce that manual work.

Minutes to Degrees

Because:

1° = 60′

the conversion is:

Degrees = Minutes ÷ 60

For example:

120′ ÷ 60 = 2°

Therefore:

120 arcminutes = 2 degrees

NIST lists one angular minute as approximately 2.908882 × 10⁻⁴ radians.

Seconds to Degrees

Because:

1° = 3,600″

the conversion is:

Degrees = Seconds ÷ 3600

For example:

7,200″ ÷ 3,600 = 2°

Therefore:

7,200 arcseconds = 2 degrees

One arcsecond is a very small angular measurement, which makes it useful for precise geographic, astronomical, and surveying measurements.

Revolutions to Degrees

A revolution represents one complete turn.

One complete circle equals:

360°

Therefore:

1 revolution = 360°

The formula is:

Degrees = Revolutions × 360

For example:

2 revolutions × 360 = 720°

And:

0.5 revolution × 360 = 180°

Common Revolution Conversions

Revolutions	Degrees	Radians
0.25	90°	π/2
0.5	180°	π
0.75	270°	3π/2
1	360°	2π
2	720°	4π

The NIST conversion table gives one revolution as approximately 6.283185 radians.

Degrees to Revolutions

The reverse formula is:

Revolutions = Degrees ÷ 360

For example:

180° ÷ 360 = 0.5 revolution

So 180 degrees represents half of a complete rotation.

Similarly:

90° ÷ 360 = 0.25 revolution

Angle Conversion Formula

The CalculatorKits converter uses radians as its base unit.

The general conversion approach is:

Result = (Input × From Factor) ÷ To Factor

This allows different angle units to be converted through a common reference system.

For the most common degree and radian conversion:

Degrees to radians

rad = degrees × π ÷ 180

Radians to degrees

degrees = radians × 180 ÷ π

For gradians:

1 grad = 0.9°

For revolutions:

1 revolution = 360°

For angular minutes and seconds:

1° = 60′ = 3,600″

These relationships cover most everyday angle conversions.

Common Angle Conversion Table

Unit	Equivalent in degrees	Equivalent in radians
1 degree	1°	0.0174533 rad
1 radian	57.2958°	1 rad
1 gradian	0.9°	0.015708 rad
1 arcminute	0.0166667°	0.000290888 rad
1 arcsecond	0.000277778°	0.00000484814 rad
1 revolution	360°	6.28319 rad
1 quadrant	90°	1.5708 rad

The standard relationships in this table are consistent with NIST’s published angle conversion factors.

Why Radians Matter in Mathematics

Degrees are intuitive for everyday geometry, but radians are particularly important in advanced mathematics.

Trigonometric formulas, calculus, and many physics equations naturally use radians.

For example:

sin(π/2) = 1

The same angle can be written as:

sin(90°) = 1

Both represent the same angle, but mathematical formulas involving derivatives and integrals generally assume radian measure.

For example:

d/dx sin(x) = cos(x)

is directly true when x is measured in radians.

This is one reason students often need to convert degrees into radians before using an angle in a calculus or physics formula.

Angle Conversion in Geometry

Angle conversion appears throughout geometry.

Some important reference angles are:

Angle	Type
0°	Zero angle
45°	Acute angle
90°	Right angle
180°	Straight angle
270°	Reflex angle
360°	Full rotation

For example, a right angle is:

90° = π/2 rad = 100 grad = 0.25 revolution

A straight angle is:

180° = π rad = 200 grad = 0.5 revolution

A complete rotation is:

360° = 2π rad = 400 grad = 1 revolution

These equivalences are useful when checking whether a conversion makes sense.

Positive and Negative Angles

Angles can also be positive or negative.

In the conventional mathematical coordinate system, positive angles are generally measured counterclockwise from the reference direction, while negative angles represent clockwise rotation.

For example:

90°

represents a quarter turn counterclockwise.

−90°

represents a quarter turn clockwise.

A negative angle can be converted using exactly the same conversion formulas.

For example:

−90° × π ÷ 180 = −π/2 rad

Therefore:

−90° = −1.5708 rad

The sign indicates direction, while the conversion factor changes the unit.

Angles Greater Than 360 Degrees

Angles do not have to stop at 360 degrees.

An angle of:

450°

represents:

360° + 90°

So it is equivalent to:

90°

after one complete rotation.

In radians:

450° × π ÷ 180 = 2.5π rad

For applications involving rotation, the original 450 degree value may still be important because it records the number of rotations completed.

Common Angle Conversion Mistakes

1. Confusing degrees and radians

This is one of the most common mistakes in mathematics.

Remember:

180° = π rad

not 180 radians.

2. Using 3.14 too early

π is an irrational number. If you use only 3.14 at the beginning of a calculation, you may introduce unnecessary rounding error.

For accurate work, keep π in symbolic form as long as possible.

3. Mixing arcminutes with time minutes

An angular minute and a minute of time use similar names but represent different things.

An arcminute is an angular measurement:

1′ = 1/60 degree

It is not a unit of elapsed time.

4. Confusing arcseconds with seconds of time

Similarly, an arcsecond is an angular unit:

1″ = 1/60 arcminute

It should not automatically be interpreted as a time measurement.

5. Treating gradians as degrees

A complete circle contains 400 gradians but 360 degrees.

Therefore:

1 grad = 0.9°

not 1°.

6. Forgetting that a revolution is a complete circle

One revolution equals:

360° = 2π rad

7. Rounding too early

If a result will be used in another calculation, keep several decimal places and round at the end.

How Accurate Is the CalculatorKits Angle Converter?

The CalculatorKits Angle Converter states that it uses official SI constants, performs calculations in the browser, and delays rounding until the displayed result. The page also states that its calculations use IEEE 754 floating point arithmetic and support high decimal precision.

For education, everyday geometry, mathematics, and most technical calculations, this provides a convenient way to obtain consistent results.

For certified surveying, regulated measurements, or safety critical engineering, conversion results should be checked against the relevant official standards and measurement procedures. The CalculatorKits page itself identifies the tool as informational rather than a replacement for certified measurement.

Where Are Angle Conversions Used?

Angle conversion is useful in many different fields.

Mathematics

Students frequently convert between degrees and radians when studying trigonometry, calculus, and geometry.

Physics

Radians are widely used in rotational motion, oscillations, angular velocity, and mathematical formulas.

Engineering

Engineers may work with degrees, radians, gradians, or specialized angular units depending on the application.

Surveying

Surveyors may use degrees, minutes, seconds, or gradians when working with directions and measurements.

Navigation

Bearings and geographic coordinates often require precise angular measurements.

Astronomy

Very small angles such as arcminutes and arcseconds are useful when describing positions and apparent sizes in the sky.

Mechanical systems

Rotational systems often describe shaft movement using degrees, radians, or revolutions.

Four Related CalculatorKits Tools

If your calculation involves angles together with other measurements, these four CalculatorKits tools can be useful:

1. Length Converter for converting meters, kilometers, feet, inches, miles, and other length units.
2. Speed Converter for converting km/h, mph, m/s, knots, ft/s, and other speed measurements.
3. Area Converter for converting square meters, square feet, acres, hectares, and other area units.
4. Power Converter for converting watts, kilowatts, horsepower, BTU per hour, and other power units used in engineering.

Frequently Asked Questions

What is the easiest way to convert degrees to radians?

Multiply the degree value by π and divide by 180.

Radians = Degrees × π ÷ 180

For example:

90° = π/2 rad ≈ 1.5708 rad

You can also enter the value directly into the CalculatorKits Angle Converter.

How many radians are in 180 degrees?

180° = π radians

Approximately:

180° = 3.14159 radians

How many degrees are in one radian?

1 radian ≈ 57.2958°

How many degrees are in a full circle?

A complete circle contains:

360°

It is also equal to:

2π radians

400 gradians

1 revolution

How many gradians are in 90 degrees?

90° = 100 gradians

How many minutes are in one degree?

1 degree = 60 arcminutes

How many seconds are in one degree?

1 degree = 3,600 arcseconds

What is DMS in angle measurement?

DMS means Degrees, Minutes, Seconds.

One degree contains 60 minutes, and one minute contains 60 seconds.

What is the SI unit of angle?

The coherent SI unit for plane angle is the radian, with the symbol rad.

Why are radians used in calculus?

Radians provide the natural relationship between an angle and the arc length of a circle. They also make many calculus and trigonometric formulas take their standard mathematical form.

Can I convert negative angles?

Yes. Negative angles can be converted using the same formulas as positive angles.

For example:

−180° = −π radians

Can an angle be greater than 360 degrees?

Yes. Angles can represent multiple rotations.

For example:

720° = 2 complete revolutions

Is a gradian the same as a gon?

Yes. Gradian and gon are names used for the same angular unit.

A full circle contains 400 gons or gradians.

What is the difference between an angle and a solid angle?

A plane angle describes an opening between two lines or rays and is measured in radians.

A solid angle describes a region in three dimensional space and is measured in steradians.

The BIPM and NIST distinguish the radian for plane angle from the steradian for solid angle.

Is the CalculatorKits Angle Converter free?

Yes. The tool is available for free online use and does not require registration.

Does the Angle Converter work on mobile devices?

Yes. Because the converter runs in a web browser, it can be used on modern desktop and mobile browsers.

Does the Angle Converter store my calculations?

The CalculatorKits page states that calculations are performed locally in the browser and that no conversion data is uploaded or stored.

Key Takeaways

Angle conversion changes the unit used to express an angle without changing the underlying angle itself.

The most important relationships to remember are:

180° = π radians

360° = 2π radians

1 radian ≈ 57.2958°

1 gradian = 0.9°

1 revolution = 360°

1° = 60′

1° = 3,600″

For quick conversions, the CalculatorKits Angle Converter eliminates the need to repeatedly calculate these relationships by hand.

Degrees are convenient for everyday geometry and navigation. Radians are fundamental in mathematics, physics, and calculus. Gradians can be useful in surveying and engineering, while minutes and seconds provide finer angular measurements for navigation, geography, and astronomy.

When accuracy matters, keep exact values such as π for as long as practical, avoid rounding too early, and verify critical calculations against the applicable standard.

References

1. NIST Guide to the SI, Appendix B.9: Angle Conversion Factors
2. NIST Special Publication 330, Section 5: Plane Angles and Solid Angles
3. BIPM International System of Units, SI Brochure
4. BIPM International Vocabulary of Metrology

Written and reviewed by the CalculatorKits Editorial Team
Last Updated: September 10, 2026