# 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

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: