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

radian in circle

1 radian is about 57.2958 degrees  

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

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
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.

Small Angles

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

x (radians) sin(x) tan(x)
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:

sine of small 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

We also know that 1 radian is about 57 degrees, so 0.01 radians is about 0.57 degrees

Also the cosine function gets close to 1 for small radian values.


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