Angle Measurement

Units of Angle Measure

Degrees

A full rotation is divided into 360 equal parts.

$1 \text{ full rotation} = 360°$

$1 \text{ right angle} = 90°$

$1 \text{ straight angle} = 180°$

Radians

The ratio of arc length to radius.

$1 \text{ radian} = \text{angle where arc length = radius}$

$1 \text{ full rotation} = 2\pi \text{ radians}$

Degree-Radian Conversions

Degrees to Radians

$\text{radians} = \text{degrees} \times \frac{\pi}{180}$

Radians to Degrees

$\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Common Conversions

Degrees	Radians
$0°$	$0$
$30°$	$\frac{\pi}{6}$
$45°$	$\frac{\pi}{4}$
$60°$	$\frac{\pi}{3}$
$90°$	$\frac{\pi}{2}$
$120°$	$\frac{2\pi}{3}$
$135°$	$\frac{3\pi}{4}$
$150°$	$\frac{5\pi}{6}$
$180°$	$\pi$
$210°$	$\frac{7\pi}{6}$
$225°$	$\frac{5\pi}{4}$
$240°$	$\frac{4\pi}{3}$
$270°$	$\frac{3\pi}{2}$
$300°$	$\frac{5\pi}{3}$
$315°$	$\frac{7\pi}{4}$
$330°$	$\frac{11\pi}{6}$
$360°$	$2\pi$

Degrees, Minutes, Seconds (DMS)

Subdivisions

$1° = 60' \text{ (minutes)}$

$1' = 60'' \text{ (seconds)}$

Converting DMS to Decimal Degrees

$\text{Decimal} = \text{degrees} + \frac{\text{minutes}}{60} + \frac{\text{seconds}}{3600}$

Example: $45° 30' 36''$

$= 45 + \frac{30}{60} + \frac{36}{3600} = 45 + 0.5 + 0.01 = 45.51°$

Converting Decimal to DMS

1. Whole number = degrees
2. Multiply decimal by 60 = minutes
3. Multiply remaining decimal by 60 = seconds

Example: $72.425°$

• Degrees: $72$
• $0.425 \times 60 = 25.5$ → Minutes: $25$
• $0.5 \times 60 = 30$ → Seconds: $30$
• Result: $72° 25' 30''$

Gradians (Grads)

Used in some surveying applications.

$1 \text{ full rotation} = 400 \text{ gradians}$

$1 \text{ right angle} = 100 \text{ gradians}$

Conversion

$\text{gradians} = \text{degrees} \times \frac{10}{9}$

$\text{degrees} = \text{gradians} \times \frac{9}{10}$

Arc Length

The length of an arc of a circle:

$s = r\theta \quad (\theta \text{ in radians})$

$s = \frac{\theta}{360} \times 2\pi r \quad (\theta \text{ in degrees})$

Where:

• $s$ = arc length
• $r$ = radius
• $\theta$ = central angle

Sector Area

The area of a sector of a circle:

$A = \frac{1}{2}r^2\theta \quad (\theta \text{ in radians})$

$A = \frac{\theta}{360} \times \pi r^2 \quad (\theta \text{ in degrees})$

Angular Speed

$\omega = \frac{\theta}{t} \quad \text{(radians per unit time)}$

Linear Speed

For a point on a rotating object:

$v = r\omega$

Where:

• $v$ = linear speed
• $r$ = radius
• $\omega$ = angular speed (rad/time)

Coterminal Angles

Angles that share the same terminal side.

$\theta \pm 360°n \quad \text{(degrees)}$

$\theta \pm 2\pi n \quad \text{(radians)}$

Where $n$ is any integer.

Finding Coterminal Angle in $[0°, 360°)$

Add or subtract $360°$ until the angle is in range.

Example: $-150°$

$-150° + 360° = 210°$

Reference Angles

The acute angle formed with the x-axis.

Quadrant	Reference Angle (degrees)	Reference Angle (radians)
I	$\theta' = \theta$	$\theta' = \theta$
II	$\theta' = 180° - \theta$	$\theta' = \pi - \theta$
III	$\theta' = \theta - 180°$	$\theta' = \theta - \pi$
IV	$\theta' = 360° - \theta$	$\theta' = 2\pi - \theta$