Degrees to Radians (and Radians to Degrees): Formula and Table

The short answer

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. So 90° = 90 × π/180 = π/2 ≈ 1.5708 rad, and 1 rad ≈ 57.2958°. The Angle Converter does this instantly and also shows the angle in gradians, arcminutes, arcseconds and turns.

Why there are two units

A full circle is 360 degrees or 2π radians. Both describe the same turn; they just divide it differently.

• Degrees split a circle into 360 equal parts. They are easy to picture and are used in everyday life, geometry classes, maps and construction.
• Radians measure an angle by arc length: one radian is the angle at the centre of a circle when the arc is as long as the radius. Calculus, physics and almost all programming languages use radians, because formulas such as the derivative of sin x only work neatly in radians.

Because 360° = 2π rad, 180° = π rad. Every conversion comes from that one fact.

The formulas

• radians = degrees × π ÷ 180
• degrees = radians × 180 ÷ π

Useful constants:

• 1° = 0.01745329 rad
• 1 rad = 57.29577951°

Common angles: degrees to radians

Degrees	Radians (exact)	Radians (decimal)
1°	π/180	0.01745329
15°	π/12	0.26179939
30°	π/6	0.52359878
45°	π/4	0.78539816
60°	π/3	1.04719755
90°	π/2	1.57079633
120°	2π/3	2.0943951
135°	3π/4	2.35619449
180°	π	3.14159265
270°	3π/2	4.71238898
360°	2π	6.28318531

In maths classes, answers are usually left as fractions of π (for example π/3, not 1.0472). Divide the degrees by 180, simplify the fraction, and attach π: 120/180 = 2/3, so 120° = 2π/3.

Radians to degrees

Radians	Degrees
0.5	28.64788976°
1	57.29577951°
2	114.59155903°
2.5	143.23944878°
π/3	60°

For a value written with π, replace π with 180°: 5π/6 = 5 × 180 ÷ 6 = 150°.

How to use the Angle Converter

Open the Angle Converter.

Type the angle in Value to convert.

Choose the unit you have in From: Degree (°) or Radian (rad).

Pick the unit you want in the result menu. Degrees to radians is the default pair.

The All Angle conversions list shows the same angle in degrees, radians, gradians, arcminutes, arcseconds and turns. Click a row to copy it.

Results are shown to up to 8 decimal places, which is more precision than most homework or engineering drawings need.

Worked problem: arc length and sector area

Radians make circle formulas simple, which is the main reason they are used in maths and physics. With the angle θ in radians:

• Arc length = r × θ
• Sector area = ½ × r² × θ

Example: a circle has radius 10 cm. Find the arc length and sector area for a 60° angle.

Convert 60° to radians: 60 × π/180 = π/3 = 1.04719755 rad.

Arc length = 10 × 1.04719755 = 10.47 cm.

Sector area = ½ × 10² × 1.04719755 = 52.36 cm².

If you plug 60 straight into these formulas without converting, you get an arc of 600 cm, which is clearly impossible for a circle with a 10 cm radius. That kind of check catches most degree and radian slips.

Degrees, minutes and seconds (DMS)

Surveying, astronomy and map coordinates often write angles as degrees, minutes and seconds, for example 12° 30′ 36″. One degree = 60 arcminutes and one arcminute = 60 arcseconds.

To convert DMS to radians:

Turn it into decimal degrees: 12 + 30 ÷ 60 + 36 ÷ 3600 = 12.51°.

Enter 12.51 in the Angle Converter with Degree as the From unit.

Read off radians: 0.21834069 rad.

The converter's Arcminute and Arcsecond units help with the parts. For example, 0.51° is 1,836 arcseconds, and 0.2° is 12 arcminutes.

Gradians and turns

• Gradians (also called gon) divide a circle into 400 parts, so 90° = 100 grad. They appear on some surveying instruments and as the GRAD mode on some scientific calculators.
• Turns count whole rotations: 1 turn = 360°, so 0.25 turn = 90°.

Radians in spreadsheets and code

Many tools expect radians even when you think in degrees:

• Excel and Google Sheets: SIN(), COS() and TAN() take radians. Use RADIANS(30) to convert, or DEGREES() for the reverse.
• JavaScript: Math.sin() uses radians. Convert with deg * Math.PI / 180.
• Python: math.sin() uses radians. Use math.radians(30) and math.degrees(x).

If sin(30) comes out as −0.988 instead of 0.5, the function treated 30 as radians. Convert first.

Common mistakes

• Flipping the fraction. Degrees to radians multiplies by π/180 (the result gets smaller). Radians to degrees multiplies by 180/π (the result gets bigger). If 90° turned into 5,156, you used the wrong one.
• Using 3.14 for π in precise work. 180° × 3.14/180 = 3.14, but the true value is 3.14159265. Small differences grow when angles are used in further calculations.
• Typing DMS as a decimal. 12° 30′ is 12.5°, not 12.30°.
• Mixing up the calculator mode. On a scientific calculator the DEG/RAD setting changes how trig functions read your input. See Degrees vs Radians: Why Your Calculator Gives the Wrong sin Answer and try it on the Scientific Calculator.