Chapter 1

The Units of Arc and Angle

Definition of Angle:  Angle is defined as the space between two crossing lines, as shown below:

We may call an angle a "central angle" if its vertex is at the center of a circle, as shown.  The three angles shown are central angles.  Anglewise, the size of any central angle is equal to its opposite arc as shown below:

Three different units are defined for the measurement of arc or angle: degree, grad, and radian.

1) Degree

1/360 of a circle is defined as 1° called "one degree.

1/60 of a degree is 1 minute, and 1/60 of a minute is 1 second of angle.

1/400 of a circle is defined as 1 grad.

One radian of angle is the central angle in any circle which opposite arc is equal to the radius of that circle.  If a piece of string is cut equal to the radius R of a circle and then placed on the edge of that circle, as shown, the central angle corresponding (or opposite) to that arc is called one "radian."

Example 1:  Naming 3.14rd as " π ", calculate angles 360°, 180°, 90°, 60°, 45°, and 30° in terms of π. The pronunciation of the symbol is "pi."

Solution:

 Fraction: 1 full circle 1/2 circle 1/4 circle 1/6 circle 1/8 circle 1/12 circle Degrees: 360° 180° 90° 60° 45° 30° Radians: 2π = 6.28rd π π/2 π/3 π/4 π/6

Conversion Between Units of Angle:

To convert from Degrees D to grad G or to Radians rd, the following relations apply:

Solution: According to the above formula, we may write: (18/180) = (G/200) = (rd/3.14).

From the first two ratios: G = (200X18) / 180 = 20 grads.

From the 1st and 3rd ratios: rd = (18X3.14) / 180 = 0.314 radians.

Solution: According to the above formula, we may write: (D/180) = (15/200) = (rd/3.14).

From the first two ratios: D = (180X15) / 200 = 13.5 degrees.

From the last two ratios: rd = (15X3.14) / 200 = 0.236 radians.

Solution: According to the above formula, we may write: (D/180) = (G/200) =  / 7) /π.

From the 1st and 3rd ratios: D = 180X / 7) / π  = 180 / 7  = 25.7 degrees.

From the 2nd and 3rd ratios: G = 200X / 7) / π  = 200 / 7  = 28.6 grads.

Arc Length-Central Angle Formula:

There is an easy formula that relates any central angle ( θ ) to its opposite arclength ( s ) and the radius ( R ) of a circle.  This formula is valid only if the central angle is measured or expressed in radians.

Example 5:  Referring to the above figure, suppose angle θ is 148° and R = 1.25 in.  Calculate the length of arc AB.

Solution: Using S = R θ, and converting degrees to radians, yields:    S = (1.25 in.)(148°)( 3.14rd / 180° ) = 3.2 in.

Exercises:

1) Convert the following angles from radians to degrees and grads:  a) π/2   b)π   c) 3π/2.