Angles in Standard Position (With Examples)

An angle in standard position has its vertex at the origin and its initial side on the positive x-axis. The terminal side is where the rotation stops. Turning counterclockwise makes a positive angle, and turning clockwise makes a negative angle. The quadrant where the terminal side lands helps describe the angle.

Before this lesson: You should know Measuring and Classifying Angles, the Coordinate Plane and Trigonometric Ratios.

Angles as Rotations

In geometry, an angle is two rays that share an endpoint, and its measure is between 0° and 180°. In trigonometry, an angle is a rotation. One ray stays put, and the other turns around the vertex. Because a ray can keep turning, an angle can be bigger than 360° or even negative.

Putting every angle in the same place makes them easy to compare. That place is called standard position:

The vertex is at the origin, $(0, 0)$ .
The initial side lies along the positive x-axis.
The terminal side is the ray after the rotation.
A positive angle turns counterclockwise. A negative angle turns clockwise.

A positive angle, like 135°, turns counterclockwise. A negative angle, like −60°, turns clockwise.

Quadrants and Quadrantal Angles

Greek letters such as $\theta$ (theta) are often used to name angles. The quadrant that holds the terminal side depends on the size of $\theta$ :

Terminal side in	Positive angles	Negative angles
Quadrant I	$0^\circ \lt \theta \lt 90^\circ$	$-360^\circ \lt \theta \lt -270^\circ$
Quadrant II	$90^\circ \lt \theta \lt 180^\circ$	$-270^\circ \lt \theta \lt -180^\circ$
Quadrant III	$180^\circ \lt \theta \lt 270^\circ$	$-180^\circ \lt \theta \lt -90^\circ$
Quadrant IV	$270^\circ \lt \theta \lt 360^\circ$	$-90^\circ \lt \theta \lt 0^\circ$

An angle whose terminal side lies on an axis, such as 0°, 90°, 180°, 270° or −90°, is a quadrantal angle. It is not in any quadrant.

Draw the initial side on the positive x-axis.
If the angle is more than 360° or less than −360°, add or subtract 360° until it is between −360° and 360°. Each 360° is one full turn.
Turn counterclockwise for a positive angle or clockwise for a negative angle.
Use the table to name the quadrant, or the axis for a quadrantal angle.

Worked Examples

Example 1: A positive angle

Draw $135^\circ$ in standard position and name its quadrant.

Start with the initial side on the positive x-axis.

The angle is positive, so turn counterclockwise.

90^\circ \lt 135^\circ \lt 180^\circ

Compare with the quadrant borders.

135^\circ - 90^\circ = 45^\circ

Find how far past 90° it goes.

Draw the terminal side 45° past the positive y-axis.

Answer: The terminal side is in Quadrant II, halfway between the positive y-axis and the negative x-axis.

Example 2: A negative angle

Draw $-60^\circ$ in standard position and name its quadrant.

Start with the initial side on the positive x-axis.

The angle is negative, so turn clockwise.

-90^\circ \lt -60^\circ \lt 0^\circ

Compare with the quadrant borders.

Draw the terminal side 60° below the positive x-axis.

Answer: The terminal side is in Quadrant IV.

Example 3: An angle greater than 360°

Name the quadrant of $500^\circ$ .

The angle is more than 360°, so it makes one full turn first.

500^\circ - 360^\circ = 140^\circ

Subtract one full turn.

90^\circ \lt 140^\circ \lt 180^\circ

Compare with the quadrant borders.

Answer: Quadrant II. The angle turns once all the way around, then 140° more.

Example 4: A large negative angle

Name the quadrant of $-250^\circ$ .

The angle is negative, so it turns clockwise.

-270^\circ \lt -250^\circ \lt -180^\circ

Compare with the borders for negative angles.

Check by adding one full turn: $-250^\circ + 360^\circ = 110^\circ$

Compare: $90^\circ \lt 110^\circ \lt 180^\circ$ , which is also Quadrant II

Answer: Quadrant II. Turning 250° clockwise ends at the same place as turning 110° counterclockwise.

Example 5: Word problem with a skateboard spin

A skateboarder faces east, along the positive x-axis, and does a “540,” a counterclockwise spin of 540°. Which way does the skateboarder face when the spin ends?

540^\circ - 360^\circ = 180^\circ

The spin is more than 360°, so subtract one full turn.

An angle of 180° is a quadrantal angle. Its terminal side is the negative x-axis.

The negative x-axis points west.

Answer: The skateboarder ends facing west, the opposite direction, after one and a half turns.

Common Mistakes

Mistake	Example of the error	Fix
Turning the wrong way	Drawing −60° counterclockwise into Quadrant I	Negative angles turn clockwise, so −60° is in Quadrant IV.
Starting from the wrong side	Measuring 135° from the positive y-axis	The initial side is always the positive x-axis.
Mixing up the quadrant numbers	Calling the top left quadrant Quadrant IV	Quadrants are numbered I to IV counterclockwise, starting at the top right.
Putting a quadrantal angle in a quadrant	Saying 270° is in Quadrant III	270° lies on the negative y-axis, between Quadrants III and IV.
Forgetting full turns	Saying 500° is not a real angle	Subtract 360°: 500° ends where 140° ends.

Related Methods

Degrees, minutes and seconds

A degree can be split into 60 minutes (′), and a minute into 60 seconds (″). To change $25.75^\circ$ , multiply the decimal part by 60: $0.75 \cdot 60 = 45$ , so $25.75^\circ = 25^\circ 45'$ .

Angles that share a terminal side

In Example 4, −250° and 110° end on the same ray. Angles like these are called coterminal angles, and they are the topic of the next lesson.

Measuring in radians

Angles can also be measured in radians, where one full turn is $2\pi$ instead of 360°. Radians come later in this unit and are the main unit in calculus.

Practice Problems

Name the quadrant of $75^\circ$ .
Name the quadrant of $210^\circ$ .
Name the quadrant of $-45^\circ$ .
Name the quadrant of $300^\circ$ .
Name the quadrant of $-135^\circ$ .
Where is the terminal side of $180^\circ$ ?
Name the quadrant of $420^\circ$ .
Name the quadrant of $-390^\circ$ .
The terminal side of an angle lies on the negative y-axis. Give one positive angle and one negative angle between −360° and 360° that fit.
A wind turbine blade starts on the positive x-axis and turns 1,000° counterclockwise. In which quadrant does it stop?

Show answers

0^\circ \lt 75^\circ \lt 90^\circ

Compare.

Answer: Quadrant I

180^\circ \lt 210^\circ \lt 270^\circ

Compare.

Answer: Quadrant III

The angle is negative, so turn clockwise

-90^\circ \lt -45^\circ \lt 0^\circ

Compare.

Answer: Quadrant IV

270^\circ \lt 300^\circ \lt 360^\circ

Compare.

Answer: Quadrant IV

The angle is negative, so turn clockwise

-180^\circ \lt -135^\circ \lt -90^\circ

Compare.

Answer: Quadrant III

180° is half a turn, so it is a quadrantal angle
Answer: on the negative x-axis

420^\circ - 360^\circ = 60^\circ

Subtract one full turn.

0^\circ \lt 60^\circ \lt 90^\circ

Compare.

Answer: Quadrant I

-390^\circ + 360^\circ = -30^\circ

Add one full turn.

-90^\circ \lt -30^\circ \lt 0^\circ

Compare.

Answer: Quadrant IV

3 \cdot 90^\circ = 270^\circ

Counterclockwise, the negative y-axis is three quarter turns.

-90^\circ

Clockwise, it is one quarter turn.

Answer: $270^\circ$ and $-90^\circ$

10.

Find the number of full turns: $\frac{1000}{360} \approx 2.8$ , so there are 2 full turns

2 \cdot 360^\circ = 720^\circ

Find the full turns in degrees.

1000^\circ - 720^\circ = 280^\circ

Subtract.

270^\circ \lt 280^\circ \lt 360^\circ

Compare.

Answer: Quadrant IV

FAQ

What is an angle in standard position?

It is an angle with its vertex at the origin and its initial side on the positive x-axis. The terminal side shows how far, and in which direction, the angle turns.

What is the difference between the initial side and the terminal side?

The initial side is where the rotation starts, on the positive x-axis. The terminal side is where the rotation ends.

How can an angle be negative?

A negative angle turns clockwise instead of counterclockwise. For example, −90° turns a quarter turn clockwise and ends on the negative y-axis.

What is a quadrantal angle?

It is an angle whose terminal side lies on the x-axis or y-axis, such as 0°, 90°, 180° or 270°. It does not lie in any quadrant.

Summary

In standard position, the vertex is at the origin and the initial side is the positive x-axis.
Positive angles turn counterclockwise; negative angles turn clockwise.
The terminal side’s quadrant depends on the size of the angle.
Quadrantal angles, like 90° and 180°, end on an axis.
Add or subtract 360° to handle angles beyond one full turn.

Previous course: Algebra 2 Index

Next lesson: Coterminal Angles

Back to the Trigonometry Index