Degrees to Radians and Radians to Degrees: Formula, Chart

To convert degrees to radians, multiply by ( frac{pi}{180} ): ( 90^circ = 90cdotfrac{pi}{180} = frac{pi}{2} approx 1.5708 ) rad. To convert radians to degrees, multiply by ( frac{180}{pi} ): ( 1 text{ rad} = frac{180}{pi} approx 57.2958^circ ). Both come from ( 360^circ = 2pi ) radians, so ( 180^circ = pi ) radians.

Degrees and radians measure angles in different ways. Degrees split a full turn into 360 equal parts. Radians measure an angle by comparing its arc length to the radius of a circle. Higher mathematics, physics, and computer graphics rely heavily on radians because they simplify core formulas. You need quick, accurate conversions between these two systems whenever you work in geometry, trigonometry, and calculus.

The conversion formulas

$$text{radians} = text{degrees} times frac{pi}{180}$$

$$text{degrees} = text{radians} times frac{180}{pi}$$

You can remember these formulas through unit cancellation. Treat units like algebraic terms. When you convert degrees to radians, place degrees in the denominator so the degree unit cancels out, leaving radians with ( pi ) on top. When you convert radians to degrees, place ( pi ) radians in the denominator so the radian unit cancels out, leaving degrees with ( 180^circ ) on top.

How to convert degrees to radians

1. Multiply your angle in degrees by ( pi ).
2. Divide the result by 180.
3. Find the greatest common factor of your degree number and 180 to simplify the fraction.
4. Leave ( pi ) in your fraction for an exact symbolic value, or evaluate ( pi approx 3.14159 ) on a calculator for a decimal approximation.

Example 1. Convert ( 60^circ ) to radians. Multiply 60 by ( frac{pi}{180} ) to get ( frac{60pi}{180} ). Divide both the numerator and denominator by 60. You get the exact fraction ( frac{pi}{3} ) rad, which equals approximately 1.0472 rad.

Example 2. Convert ( 135^circ ) to radians. Multiply 135 by ( frac{pi}{180} ) to get ( frac{135pi}{180} ). Divide 135 and 180 by their greatest common factor, 45. You get ( frac{3pi}{4} ) rad, or roughly 2.3562 rad.

Example 3. Convert ( 75^circ ) to radians. Multiply 75 by ( frac{pi}{180} ) to yield ( frac{75pi}{180} ). Divide both numbers by 15. The simplified result is ( frac{5pi}{12} ) rad, which equals approximately 1.3090 rad.

Example 4. Convert ( -210^circ ) to radians. Multiply -210 by ( frac{pi}{180} ) to obtain ( -frac{210pi}{180} ). Divide top and bottom by 30 to get ( -frac{7pi}{6} ) rad. Keep the negative sign throughout your calculation because negative angles represent clockwise rotation.

Converting degrees, minutes, and seconds (DMS) to radians

Fields like land surveying and astronomy express angles in degrees, minutes, and seconds (DMS) rather than decimal degrees. One degree contains 60 minutes ( ( 60′ ) ), and one minute contains 60 seconds ( ( 60” ) ). You convert DMS measurements into decimal degrees before converting them to radians.

1. Divide the seconds by 3600 to turn them into decimal degrees.
2. Divide the minutes by 60 to turn them into decimal degrees.
3. Add these decimal values to the whole degree count.
4. Multiply the final decimal degree value by ( frac{pi}{180} ).

For example, to convert ( 52^circ 30′ 36” ) to radians:

First, convert seconds to degrees: ( frac{36}{3600} = 0.01^circ ).

Next, convert minutes to degrees: ( frac{30}{60} = 0.5^circ ).

Add them together: ( 52 + 0.5 + 0.01 = 52.51^circ ).

Finally, multiply by ( frac{pi}{180} ): ( 52.51 times frac{pi}{180} approx 0.9165 ) rad.

How to convert radians to degrees

When you convert from radians back to degrees, reverse the process by multiplying by ( frac{180}{pi} ).

1. Multiply the angle in radians by 180.
2. Divide the result by ( pi ).
3. Cancel out ( pi ) if it appears in the numerator.
4. Simplify the remaining fraction or decimal to find the degree measure.

Example 5. Convert ( frac{5pi}{6} ) radians to degrees. Multiply ( frac{5pi}{6} ) by ( frac{180}{pi} ). The ( pi ) terms cancel out immediately: ( frac{5pi}{6}cdotfrac{180}{pi} = 150^circ ).

Example 6. Convert ( frac{7pi}{4} ) radians to degrees. Multiply ( frac{7pi}{4} ) by ( frac{180}{pi} ). Cancel ( pi ) and divide 180 by 4 to get 45. Multiply 7 by ( 45^circ ): ( frac{7pi}{4} = frac{7cdot180^circ}{4} = 315^circ ).

Example 7: no ( pi ). Convert 2 radians to degrees. Radians do not always show ( pi ) explicitly. Multiply 2 by ( frac{180}{pi} ): ( 2 ) rad ( = 2cdotfrac{180}{pi} approx 114.59^circ ). An angle of 2.5 rad equals ( 2.5cdotfrac{180}{pi} approx 143.24^circ ).

When an angle contains ( pi ), use a quick mental shortcut: replace ( pi ) directly with ( 180^circ ) and simplify. For example, ( frac{2pi}{3} to frac{2 cdot 180^circ}{3} = 120^circ ).

Converting coterminal and negative angles

Angles can exceed ( 360^circ ) ( ( 2pi ) radians) or fall below ( 0^circ ) (0 radians). The conversion steps remain identical regardless of the size or sign of the angle.

For large angles like ( 720^circ ), multiply by ( frac{pi}{180} ) to get ( frac{720pi}{180} = 4pi ) rad. Two full revolutions equal ( 4pi ) radians. If you need the primary coterminal angle between 0 and ( 2pi ), subtract multiples of ( 2pi ) (or ( 360^circ )).

For negative angles like ( -frac{3pi}{2} ), multiply by ( frac{180}{pi} ) to get ( -frac{3}{2} times 180^circ = -270^circ ). Add ( 360^circ ) if you need a positive coterminal angle: ( -270^circ + 360^circ = 90^circ ) (which equals ( frac{pi}{2} ) rad).

Degrees to radians chart

This table lists standard angles on the unit circle in degrees, exact radian fractions, and four-decimal-place radian approximations.

Degrees	Radians (exact)	Radians (decimal)
( 0^circ )	0	0
( 30^circ )	( frac{pi}{6} )	0.5236
( 45^circ )	( frac{pi}{4} )	0.7854
( 60^circ )	( frac{pi}{3} )	1.0472
( 90^circ )	( frac{pi}{2} )	1.5708
( 120^circ )	( frac{2pi}{3} )	2.0944
( 135^circ )	( frac{3pi}{4} )	2.3562
( 150^circ )	( frac{5pi}{6} )	2.6180
( 180^circ )	( pi )	3.1416
( 210^circ )	( frac{7pi}{6} )	3.6652
( 225^circ )	( frac{5pi}{4} )	3.9270
( 240^circ )	( frac{4pi}{3} )	4.1888
( 270^circ )	( frac{3pi}{2} )	4.7124
( 300^circ )	( frac{5pi}{3} )	5.2360
( 315^circ )	( frac{7pi}{4} )	5.4978
( 330^circ )	( frac{11pi}{6} )	5.7596
( 360^circ )	( 2pi )	6.2832

Two base conversion values help when you make quick estimates on a calculator: ( 1^circ approx 0.017453 ) rad, and ( 1 ) rad ( approx 57.2958^circ ).

What is a radian?

Draw a circle of radius ( r ) and mark an arc whose length equals the radius. The angle at the center is one radian. Because the whole circumference is ( 2pi r ), a full turn contains ( 2pi ) radians, which is where ( 360^circ = 2pi ) comes from.

On the unit circle (radius 1), an angle of ( theta ) radians cuts off an arc of length exactly ( theta ). That makes radians the natural measure: the angle is the arc length. In general:

$$s = rtheta qquad text{and} qquad A = tfrac12r^2theta$$

for the arc length ( s ) and sector area ( A ), with ( theta ) in radians.

Example 8. A sector of a circle of radius 6 cm has angle ( 40^circ ). In radians, ( 40^circ = frac{2pi}{9} ), so the arc length is ( 6cdotfrac{2pi}{9} = frac{4pi}{3} approx 4.19 ) cm and the area is ( frac12cdot36cdotfrac{2pi}{9} = 4pi approx 12.57 ) cm².

Unit circle in radians

The standard angles across all four quadrants on the unit circle fall into three primary mathematical families based on their reduced denominator:

• Denominator 6 (( frac{pi}{6}, frac{5pi}{6}, frac{7pi}{6}, frac{11pi}{6} )): reference angle ( 30^circ ). These angles lie closest to the horizontal x-axis.
• Denominator 4 (( frac{pi}{4}, frac{3pi}{4}, frac{5pi}{4}, frac{7pi}{4} )): reference angle ( 45^circ ). These angles split each quadrant directly in half.
• Denominator 3 (( frac{pi}{3}, frac{2pi}{3}, frac{4pi}{3}, frac{5pi}{3} )): reference angle ( 60^circ ). These angles lie closest to the vertical y-axis.

These come straight from the special right triangles: the 30-60-90 triangle is the ( frac{pi}{6} )-( frac{pi}{3} )-( frac{pi}{2} ) triangle, and the 45-45-90 triangle is the ( frac{pi}{4} )-( frac{pi}{4} )-( frac{pi}{2} ) triangle.

Check a conversion

Enter an angle in degrees and the calculator shows it in radians (exact for special angles), along with its sine, cosine and tangent. Switch the unit to radians to go the other way.

The trig calculator page has more examples, and the scientific calculator has a DEG/RAD switch.

Why calculus uses radians

Derivative formulas for trig functions only work in radians. The key limit

$$lim_{xto0}frac{sin x}{x} = 1$$

is true when x is in radians; in degrees it equals ( frac{pi}{180} ). As a result, ( frac{d}{dx}sin x = cos x ) in radians, but ( frac{pi}{180}cos x ) in degrees. Every formula in the derivative of sin x and limit of sin x / x guides assumes radians, and so does the small-angle approximation ( sin x approx x ).

Who uses degrees versus radians?

Choosing between degrees and radians depends on your practical field:

Use degrees if you work in:

• Architecture and Construction: Builders measure cut angles and roof pitches in degrees for direct use with physical tools.
• Navigation and Aviation: Pilots and navigators steer using 360-degree compass headings.
• Land Surveying: Surveyors track boundaries in degrees, minutes, and seconds.

Use radians if you work in:

• Calculus and Higher Mathematics: Radians remove awkward scaling constants from derivatives and Taylor series.
• Physics and Engineering: Equations for rotational kinematics, wave propagation, and alternating current require radians.
• Computer Graphics: Game engines like Unity calculate rotational movement using radians internally.

Common mistakes

• Using the wrong factor. Going to radians, multiply by ( frac{pi}{180} ), not ( frac{180}{pi} ). A quick check: radian values of ordinary angles are small numbers (under about 6.3). If your converted degree angle gives thousands, you flipped the fraction.
• Calculator mode. ( sin(30) ) in radian mode is about ( -0.988 ), not 0.5. Always verify whether “DEG” or “RAD” displays on your calculator screen before typing trigonometric expressions.
• Not simplifying. Writing ( frac{90pi}{180} ) can cost points on exams. Divide out the greatest common factor to reduce it to ( frac{pi}{2} ).
• Dropping ( pi ). ( 180^circ = pi ), not 180 radians. Remember that 180 radians equals over 28 complete turns around a circle.

Practice problems

Convert to radians: (1) ( 20^circ ); (2) ( 240^circ ); (3) ( 405^circ ). Convert to degrees: (4) ( frac{pi}{9} ); (5) ( frac{11pi}{6} ); (6) ( 3 ) rad.

Step-by-step solutions:

(1) ( 20^circ times frac{pi}{180} = frac{20pi}{180} = frac{pi}{9} ) rad.

(2) ( 240^circ times frac{pi}{180} = frac{240pi}{180} = frac{4pi}{3} ) rad.

(3) ( 405^circ times frac{pi}{180} = frac{405pi}{180} = frac{9pi}{4} ) rad.

(4) ( frac{pi}{9} times frac{180}{pi} = frac{180}{9} = 20^circ ).

(5) ( frac{11pi}{6} times frac{180}{pi} = 11 times 30^circ = 330^circ ).

(6) ( 3 times frac{180}{pi} = frac{540}{pi} approx 171.89^circ ).

Frequently asked questions

How do you convert degrees to radians?

Multiply by ( frac{pi}{180} ) and simplify. For example, ( 45^circ = frac{45pi}{180} = frac{pi}{4} ).

How do you convert radians to degrees?

Multiply by ( frac{180}{pi} ). For example, ( frac{pi}{3} cdot frac{180}{pi} = 60^circ ).

How many degrees is 1 radian?

About ( 57.2958^circ ), which is ( frac{180}{pi} ).

How many radians are in a full circle?

( 2pi approx 6.2832 ) radians, equal to ( 360^circ ).

Why are radians used instead of degrees?

A radian links an angle directly to arc length (( s = rtheta )), which makes calculus formulas like ( frac{d}{dx}sin x = cos x ) work without extra conversion factors.

Further reading

• Radian (Wikipedia) — definition, history and conversion between units.
• Calculus I Review chapter (Paul’s Online Math Notes) — radian measure and the trig facts used in calculus.