is wrapped round the 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:

and wrap it round the circle.

Let us see why 1 Radian is equal to 57.2958... degrees: In a half circle there are π radians, which is also 180°

=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:

(exact)
(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 ### 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?

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.

## Small Angles

For small angles the values of the sine and tangent functions get close to the value of the angle in radian:

1 0.8414710 1.55740772
0.1 0.0998334 0.1003347
0.01 0.0099998 0.0100003

Here we can see the tan function on a triangle for smaller and smaller angles: At 0.01 radians both sin and tan are within 0.003% of the radian value.

### Example: A road rises 1 m for every 100 m along. What is it's angle in degrees (without using a calculator)?

1 in 100 is 0.01, and tan(0.01) is approximately 0.01 radians