Radians

The angle made when the radius
is wrapped round the circle:

radian in circle

1 Radian is about 57.2958 degrees.

Why "57.2958..." degrees? We will see in a moment.

The Radian is a pure measure based on the Radius of the circle:

Radian: the angle made when we take the radius
and wrap it round the circle.

Radians and Degrees

Let us see why 1 Radian is equal to 57.2958... degrees:

radius of 1, pi is half of circumference

In a half circle there are π radians, which is also 180°

π radians =180°

So 1 radian = 180°/π

=57.2958...°

(approximately)

To go from radians to degrees: multiply by 180, divide by π

To go from degrees to radians: multiply by π, divide by 180

Here is a table of equivalent values:

Degrees	Radians (exact)	Radians (approx)
30°	π/6	0.524
45°	π/4	0.785
60°	π/3	1.047
90°	π/2	1.571
180°	π	3.142
270°	3π/2	4.712
360°	2π	6.283

radians circle 6.28... pieces of string

Example: How Many Radians in a Full Circle?

Imagine you cut pieces of string exactly the length from the center to the circumference of a circle ...

... how many pieces do you need to go once around the circle?

Answer: 2π (or about 6.283 pieces of string).

Radians Preferred by Mathematicians

Because the radian is based on the pure idea of "the radius being laid along the circumference", it often gives simple and natural results when used in mathematics.

For example, look at the sine function for very small values:

x (radians)	1	0.1	0.01	0.001
sin(x)	0.8414710	0.0998334	0.0099998	0.0009999998

For very small values. "x" and "sin(x)" are almost the same
(as long as "x" is in Radians!)

There will be other examples like that as you learn more about mathematics.

Conclusion

So, degrees are easier to use in everyday work, but radians are much better for mathematics.