Section 6.1 Angle Measure
An angle AOB consists of two rays R
with a common vertex O (see the Figures
below). We often interpret an angle as a rotation of the ray R
. In this case, R
called the initial side, and R
is called the terminal side of the angle. If the rotation is
counterclockwise, the angle is considered positive, and if the rotation is clockwise, the angle
is considered negative.
The measure of an angle is the amount of rotation about the vertex required to move R
. Intuitively, this is how much the angle “opens.” One unit of measurement for angles
is the degree. An angle of measure 1 degree is formed by rotating the initial side
complete revolution. In calculus and other branches of mathematics, a more natural method
of measuring angles is used — radian measure. The amount an angle opens is measured along
the arc of a circle of radius 1 with its center at the vertex of the angle.
The circumference of the circle of radius 1 is 2π and so a complete revolution has measure 2π
rad, a straight angle has measure π rad, and a right angle has measure π/2 rad. An angle that
is subtended by an arc of length 2 along the unit circle has radian measure 2 (see the Figures
Since a complete revolution measured in degrees is 360
and measured in radians is 2π rad, we
get the following simple relationship between these two methods of angle measurement.