Degrees to Radians and Radians to Degrees

degrees to radians formula

rad = degrees × π ÷ 180

1 degree = 0.0174532925 radians, so multiply degrees by 0.0174532925. Going the other way, 1 radian = 57.29577951 degrees.

degrees to radians table

How many radians in degrees?

1 ° = 0.0174533 rad. There are 0.0174533 radians in 1 degrees.

30 ° = 0.523599 rad. There are 0.523599 radians in 30 degrees.

45 ° = 0.785398 rad. There are 0.785398 radians in 45 degrees.

90 ° = 1.5708 rad. There are 1.5708 radians in 90 degrees.

180 ° = 3.14159 rad. There are 3.14159 radians in 180 degrees.

radians to degrees formula

degrees = rad × 180 ÷ π

1 radian = 57.29577951 degrees, so multiply radians by 57.29577951, or divide by 0.0174532925.

radians to degrees table

How many degrees in radians?

1 rad = 57.2958 °. There are 57.2958 degrees in 1 radians.

2 rad = 114.592 °. There are 114.592 degrees in 2 radians.

3 rad = 171.887 °. There are 171.887 degrees in 3 radians.

3.14159 rad = 180 °. There are 180 degrees in 3.14159 radians.

6.28319 rad = 360 °. There are 360 degrees in 6.28319 radians.

How the Degrees To Radians works

Degrees split a full turn into 360 parts, a choice that goes back to Babylonian astronomy. Radians measure the same turn by the circle itself: one radian is the angle that cuts off an arc exactly as long as the radius. A full turn is 2π radians, about 6.283, so 360° = 2π rad and 180° = π rad.

The degrees to radians formula

Radians = degrees × π ÷ 180, and degrees = radians × 180 ÷ π. One degree is about 0.017453 rad, and one radian is about 57.2958°. To get an exact answer, write the degrees over 180 as a fraction, simplify it and put π on top: 135/180 = 3/4, so 135° = 3π/4.

Worked examples

The first example is 60°. 60 ÷ 180 = 1/3, so 60° = π/3 ≈ 1.0472 rad.

The second example goes the other way. A wheel turns through 2.5 radians, and 2.5 × 180 ÷ π ≈ 143.24°.

How many radians in a degree, and common angles

There is π/180 ≈ 0.01745 of a radian in one degree, and 57.2958 degrees in one radian. The table lists the angles you meet most often with the exact π form and the decimal.

Common angles in degrees with their radian values

Tips and pitfalls

• Check calculator mode. sin 30 is 0.5 in degree mode but −0.988 in radian mode. Most wrong trig answers come from this.
• Keep π when you can. π/3 is exact; 1.0472 is rounded. Exams usually want the π form.
• Calculus needs radians. The derivative of sin x is cos x only when x is in radians.
• See it drawn. The unit circle labels every special angle in both units.

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