Angles and Their Measurement

You have been measuring angles your whole life without even thinking about it. Every time you turn a corner, read a clock, or notice that a ramp is “steep,” you are making judgments about angles. When a skateboarder lands a “180” or a basketball player makes a “360 dunk,” they are talking about angles. So if the mathematics of angles seems scary to you, know that you are not alone - but also know this: you already understand angles intuitively. Now we are going to give you the precise language to describe what you already know.

Core Concepts

What is an Angle?

At its most basic, an angle is formed when two rays (think of them as arrows that extend forever in one direction) share the same starting point. That shared starting point is called the vertex, and the two rays are called the sides of the angle.

But here is a more useful way to think about it: an angle measures rotation. Imagine standing facing north. Now turn to face east. You have just rotated through an angle of 90 degrees. Turn all the way around to face north again, and you have rotated through 360 degrees.

When we draw angles in math, we typically use standard position: the vertex sits at the origin of a coordinate plane, one ray (the initial side) points along the positive x-axis, and the other ray (the terminal side) shows where you end up after rotating.

$$\text{Angle} = \text{Amount of rotation from initial side to terminal side}$$

Positive angles rotate counterclockwise (the direction opposite to clock hands). Negative angles rotate clockwise. This is just a convention - mathematicians had to pick one direction to call positive, and counterclockwise won.

Degree Measure: Why 360?

The degree is probably the angle unit you know best. A full rotation is 360 degrees, written as $360°$.

But why 360? This goes back to the ancient Babylonians, who used a base-60 number system. They noticed that 360 has many factors - it is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, and more. This makes it easy to divide a circle into many equal parts without dealing with messy fractions. Want to split a circle into thirds? That is $120°$ each. Into quarters? $90°$ each. Into sixths? $60°$ each. All nice whole numbers.

Here are the key degree measurements to memorize:

• Full rotation: $360°$
• Half rotation (straight line): $180°$
• Quarter rotation (right angle): $90°$
• No rotation: $0°$

Minutes and Seconds of Arc

Just as an hour can be divided into minutes and seconds for more precision, a degree can be divided into minutes and seconds of arc.

• 1 degree = 60 minutes of arc (written $60’$)
• 1 minute = 60 seconds of arc (written $60’’$)

So an angle might be written as $47° 23’ 15’’$, meaning 47 degrees, 23 minutes, and 15 seconds.

This notation is commonly used in navigation, surveying, and astronomy, where precise angle measurements matter. For most trigonometry problems, you will work with decimal degrees (like $47.3875°$) or radians instead.

To convert from degrees-minutes-seconds to decimal degrees:

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

Radian Measure: The Natural Unit

Here is where things get interesting. Degrees are convenient for everyday use, but mathematicians and scientists prefer a different unit: the radian.

A radian is defined by a beautifully simple relationship: one radian is the angle that creates an arc length equal to the radius of the circle.

Picture a circle with radius $r$. If you measure an arc along the edge of the circle that is exactly $r$ units long, the angle at the center spanning that arc is exactly 1 radian.

Since the circumference of a circle is $2\pi r$, a full rotation (going all the way around the circle) is:

$$\frac{2\pi r}{r} = 2\pi \text{ radians}$$

This gives us our key conversion:

$$360° = 2\pi \text{ radians}$$

Or, simplifying:

$$180° = \pi \text{ radians}$$

Why do mathematicians love radians? Because they make formulas simpler. Many calculus formulas only work properly when angles are measured in radians. The derivative of $\sin(x)$ is $\cos(x)$ - but only if $x$ is in radians. If $x$ were in degrees, you would need an ugly conversion factor.

Common radian values to know:

• Full rotation: $2\pi$ rad
• Half rotation: $\pi$ rad
• Quarter rotation: $\frac{\pi}{2}$ rad
• One-sixth rotation: $\frac{\pi}{3}$ rad (that is $60°$)
• One-twelfth rotation: $\frac{\pi}{6}$ rad (that is $30°$)

Converting Between Degrees and Radians

Since $180° = \pi$ radians, converting between the two is straightforward multiplication.

Degrees to radians: Multiply by $\frac{\pi}{180}$

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

Radians to degrees: Multiply by $\frac{180}{\pi}$

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

Think of it this way: you are multiplying by a clever form of 1 (since $\frac{\pi}{180°} = 1$), which changes the units without changing the actual angle.

Arc Length Formula

Here is where radians really shine. If you have a circle with radius $r$ and you want to know the length of the arc cut off by an angle $\theta$ (measured in radians), the formula is elegant:

$$s = r\theta$$

That is it. Arc length equals radius times angle. No extra factors, no conversion constants. This only works when $\theta$ is in radians - that is exactly why radians are defined the way they are.

If you are given an angle in degrees, convert to radians first, or use:

$$s = r \times \theta_{\text{degrees}} \times \frac{\pi}{180}$$

Sector Area Formula

A sector is the “pizza slice” region enclosed by two radii and an arc. Its area has an equally clean formula when $\theta$ is in radians:

$$A = \frac{1}{2}r^2\theta$$

Where does this come from? The area of a full circle is $\pi r^2$, and a sector with angle $\theta$ is the fraction $\frac{\theta}{2\pi}$ of the full circle:

$$A = \frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2}r^2\theta$$

If the angle is given in degrees, use:

$$A = \frac{\theta}{360} \times \pi r^2$$

Coterminal Angles

Two angles are coterminal if they share the same terminal side when drawn in standard position. In other words, they point in the same direction even though they represent different amounts of rotation.

For example, $30°$ and $390°$ are coterminal. Rotating $30°$ counterclockwise lands you in the same place as rotating $390°$ counterclockwise (that is one full circle plus another $30°$).

Similarly, $30°$ and $-330°$ are coterminal. Rotating $30°$ counterclockwise lands you in the same spot as rotating $330°$ clockwise.

To find coterminal angles, add or subtract multiples of $360°$ (or $2\pi$ radians):

$$\theta_{\text{coterminal}} = \theta + 360°n \quad \text{(where } n \text{ is any integer)}$$

$$\theta_{\text{coterminal}} = \theta + 2\pi n \quad \text{(in radians)}$$

Reference Angles

The reference angle is the acute angle (between $0°$ and $90°$) formed between the terminal side of an angle and the x-axis.

Reference angles are incredibly useful because the trigonometric functions of any angle can be determined from the trig functions of its reference angle (with appropriate signs based on the quadrant).

To find a reference angle:

• Quadrant I ($0°$ to $90°$): The reference angle is the angle itself
• Quadrant II ($90°$ to $180°$): Reference angle $= 180° - \theta$
• Quadrant III ($180°$ to $270°$): Reference angle $= \theta - 180°$
• Quadrant IV ($270°$ to $360°$): Reference angle $= 360° - \theta$

For example, the reference angle for $150°$ is $180° - 150° = 30°$. This means $\sin(150°)$ has the same absolute value as $\sin(30°)$ - you just need to determine the sign based on the quadrant.

Notation and Terminology

Term	Meaning	Example
Vertex	The common endpoint where two rays meet	The corner of an angle
Initial side	The starting ray (lies on positive x-axis in standard position)	The “before” position
Terminal side	The ending ray after rotation	The “after” position
Degree (°)	1/360 of a full rotation	$90°$, $180°$, $45°$
Radian (rad)	Angle where arc length equals radius	$\pi$ rad $= 180°$
Coterminal angles	Different angles with the same terminal side	$30°$ and $390°$
Reference angle	Acute angle to the nearest x-axis	Reference for $150°$ is $30°$
Arc	A portion of a circle’s circumference	The curved part of a “pizza slice”
Sector	Region bounded by two radii and an arc	A “pizza slice” shape

Examples

Key Properties and Rules

Conversion Formulas

Degrees to radians: $$\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}$$

Radians to degrees: $$\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}$$

Arc Length and Sector Area (with θ in radians)

Arc length: $$s = r\theta$$

Sector area: $$A = \frac{1}{2}r^2\theta$$

Coterminal Angles

Two angles are coterminal if they differ by a multiple of $360°$ (or $2\pi$ radians):

$$\theta_1 \text{ and } \theta_2 \text{ are coterminal if } \theta_1 - \theta_2 = 360°n$$

Reference Angles by Quadrant

Quadrant	Angle Range	Reference Angle
I	$0° < \theta < 90°$	$\theta$
II	$90° < \theta < 180°$	$180° - \theta$
III	$180° < \theta < 270°$	$\theta - 180°$
IV	$270° < \theta < 360°$	$360° - \theta$

Key Angle Conversions

Degrees	Radians
$0°$	$0$
$30°$	$\frac{\pi}{6}$
$45°$	$\frac{\pi}{4}$
$60°$	$\frac{\pi}{3}$
$90°$	$\frac{\pi}{2}$
$180°$	$\pi$
$270°$	$\frac{3\pi}{2}$
$360°$	$2\pi$

Real-World Applications

Clock Hands and Time

The minute hand of a clock makes a full rotation ($360°$ or $2\pi$ radians) every 60 minutes. In 15 minutes, it rotates $\frac{15}{60} \times 360° = 90°$. The hour hand moves $360°$ in 12 hours, so it rotates $30°$ per hour. At 3:00, the angle between the hands is exactly $90°$.

Latitude and Longitude

Geographic coordinates use degrees, minutes, and seconds. New York City is at approximately $40° 42’ 46’’ N$ latitude and $74° 0’ 22’’ W$ longitude. Each degree of latitude corresponds to about 111 kilometers on Earth’s surface.

Steering Wheels and Rotation

When you turn a car’s steering wheel, you are rotating through angles. A “full turn” of the wheel is $360°$. Most cars require between 2.5 and 3.5 full rotations ($900°$ to $1260°$) to go from lock to lock (full left to full right).

Circular Motion in Physics

Objects moving in circles - satellites, merry-go-rounds, planets - are described using angular measurements. Angular velocity is measured in radians per second. A Ferris wheel rotating at $\frac{\pi}{30}$ rad/s completes one revolution in $\frac{2\pi}{\pi/30} = 60$ seconds.

Pizza Slices: Sector Area in Action

When a pizza is cut into 8 equal slices, each slice is a sector with central angle $\frac{360°}{8} = 45°$ or $\frac{\pi}{4}$ radians. If the pizza has a 14-inch diameter (7-inch radius), each slice has area:

$$A = \frac{1}{2}(7)^2 \times \frac{\pi}{4} = \frac{49\pi}{8} \approx 19.2 \text{ square inches}$$

Self-Test Problems

Summary

• An angle measures rotation from an initial side to a terminal side, with positive angles rotating counterclockwise.

• Degrees divide a full rotation into 360 equal parts. This system, inherited from the Babylonians, gives nice whole numbers for common fractions of a circle.

• Radians define angles by the relationship between arc length and radius: one radian creates an arc length equal to the radius. A full rotation is $2\pi$ radians.

• To convert degrees to radians, multiply by $\frac{\pi}{180}$. To convert radians to degrees, multiply by $\frac{180}{\pi}$.

• The arc length formula $s = r\theta$ works when $\theta$ is in radians. The sector area formula is $A = \frac{1}{2}r^2\theta$.

• Coterminal angles share the same terminal side but differ by full rotations ($360°$ or $2\pi$ radians).

• The reference angle is the acute angle formed with the x-axis, useful for evaluating trigonometric functions in any quadrant.

• Angles appear everywhere in real life: clocks, navigation, steering, physics, and even pizza slicing.