 Home > College of Sciences > Institute of Fundamental Sciences > Maths First > Online Maths Help > Trigonometry > Measuring Angles (Degrees and Radians) SEARCH MASSEY  # Trigonometry

Measurement of Angles

On a Cartesian coordinate system, we draw a circle centred at the origin and with radius 1.

Let P be a point on the circle.

The line OP and the positive x-axis form an angle

Let us agree that an angle measured counter-clockwise is positive, and an angle measured clockwise is negative. (See diagram below.)  ## Degrees

Starting from the positive x-axis, the point P travels around the circle in a counter-clockwise manner.

 When it arrives at the positive y-axis (a quarter of the way around the circle) its angle measures 90°. When it reaches the negative x-axis (and is now halfway around the circle), its angle measures 180°. When it hits the negative y-axis (and is three quarters of the way around the circle) it angle measures 270°. And when it returns to the positive x-axis (and has completed a full circle) its angle measures 360°.    If the point travels clockwise, when it hits the negative y-axis its angle will measure -90° and when it hits the negative x-axis its angle will measure -180°.  Exercise 1.

Enter the number of degrees of the angle in each case:

 P = (x,y) Angle Size Angle Size  x = 1, y = 0 x > 0, x = y x = 0, y = 1 x < 0, y = -x x = -1, y = 0 x < 0, y = x x = 0, y = -1 x > 0, y = -x

If the length of the arc from (1, 0) to P is 1unit we say the angle is 1 radian ( rad in short). When the unit of an angle is absent, we assume it is in radians. The circumference of the circle of radius 1 is 2π or 2π radians.

So, measuring counter-clockwise,

2π = 360° and π = 180°.

This provides us with the following conversion formulas for changing between degrees D and radians R: or or Exercise 2.

Fill in the blanks (type pi for π) to convert each of the following:

 Degrees Radians 2π 180 π/2 π/3 45 30 0