# Radian and degree measures of angles

*Degree and radian measures of angles.*

Relation of a circle radius and a circumference

length. Table of degree and radian measures

for some most used angles.

Relation of a circle radius and a circumference

length. Table of degree and radian measures

for some most used angles.

**
A
degree measure.
**
Here a unit of measurement is a

*degree*(its designation is ° or

*deg*)

*–*a turn of a ray by the 1 / 360 part of the one complete revolution. So, the complete revolution of a ray is equal to 360 deg. One degree is divided into 60

*minutes*(a designation is ‘ or

*min*); one minute – correspondingly into 60

*seconds*(a designation is “ or

*sec*)

*.*

**
A radian measure.
**
As we know from plane geometry (
see
the point "A length of arc" of the paragraph "Geometric locus. Circle and circumference"
), a length of an arc

*l ,*a radius

*r*and a corresponding central angle are tied by the relation:

*= l / r .*

This formula is a base for definition of a
*
radian measure
*
of angles. So, if
*
l
*
=
*
r ,
*
then
= 1, and we say, that an angle
is equal to1 radian, that is designed as
= 1
*
rad
*
. Thus, we have the following definition of a radian measure unit:

*
A radian is a central angle,
*
*
for which lengths of its arc and radius are equal
*
( A
*
m
*
B =
AO, Fig.1 ). So,
*
a radian measure of any angle is a ratio of a length
*
*
of an arc drawn by an arbitrary radius and concluded between sides of
this
*
*
angle to the arc radius.
*

Following this formula, a length of a circumference
*
C
*
and its radius
*
r
*
can be
expressed as:

*C / r .*

So
*
,
*
a round angle, equal to 360° in a degree measure, is simultaneously
*
*
2
in a radian measure. Hence, we receive a value of one radian:

Inversely,

It is useful to remember the following comparative table of degree and radian measure for some angles, we often deal with: