Quadrants

Quadrants

The four sections into which the x-y plane is divided by the x- and y-axes.

X-Y coordinate plane divided into four quadrants: Quadrant I (top right), II (top left), III (bottom left), IV (bottom right).

See also

Octants

Key Formula

\text{Quadrant I: } (x > 0,\; y > 0) \qquad \text{Quadrant II: } (x < 0,\; y > 0)

\text{Quadrant III: } (x < 0,\; y < 0) \qquad \text{Quadrant IV: } (x > 0,\; y < 0)

Where:

$x$ = The horizontal coordinate (distance left or right of the origin)
$y$ = The vertical coordinate (distance above or below the origin)

Worked Example

Problem: Determine which quadrant the point (−5, 3) lies in.

Step 1: Identify the sign of the x-coordinate. Here x = −5, which is negative.

x = -5 < 0

Step 2: Identify the sign of the y-coordinate. Here y = 3, which is positive.

y = 3 > 0

Step 3: Match the sign combination to a quadrant. A negative x and a positive y corresponds to Quadrant II (upper-left region).

x < 0 \text{ and } y > 0 \implies \text{Quadrant II}

Answer: The point (−5, 3) lies in Quadrant II.

Another Example

This example shows the common edge case where a coordinate equals zero. Points on the axes are not assigned to any quadrant.

Problem: A point has coordinates (0, −7). Which quadrant does it lie in?

Step 1: Examine the x-coordinate. Here x = 0.

x = 0

Step 2: Because x = 0, the point sits exactly on the y-axis. A point must have both coordinates nonzero to belong to a quadrant.

Step 3: Conclude that this point is on the boundary between Quadrant III and Quadrant IV, not inside any quadrant.

Answer: The point (0, −7) does not lie in any quadrant — it lies on the y-axis.

Frequently Asked Questions

Why are quadrants numbered counterclockwise?

The convention follows the standard direction of positive angle measurement in mathematics. Angles are measured counterclockwise from the positive x-axis, so the quadrants are numbered in the same direction: starting from the upper-right (Quadrant I) and rotating counterclockwise through II, III, and IV.

What quadrant is a point in if one of its coordinates is zero?

If either coordinate is zero, the point lies on an axis rather than in a quadrant. For example, (0, 5) is on the y-axis and (−3, 0) is on the x-axis. The origin (0, 0) is the intersection of both axes and also does not belong to any quadrant.

How do you remember which quadrant is which?

Start in the upper-right corner and count counterclockwise: I, II, III, IV. A helpful mnemonic for the sign patterns is 'All Students Take Calculus' — All positive in I, Sine (y) positive in II, Tangent positive in III, Cosine (x) positive in IV. This works because sine corresponds to the y-value and cosine to the x-value on the unit circle.

Quadrants (2D) vs. Octants (3D)

	Quadrants (2D)	Octants (3D)
Definition	Four regions of the x-y plane divided by two axes	Eight regions of 3D space divided by three coordinate planes
Number of regions	4 (quad = four)	8 (oct = eight)
Coordinates involved	x and y	x, y, and z
Sign combinations	(+,+), (−,+), (−,−), (+,−)	All eight combinations of ± for three coordinates
When encountered	Algebra, trigonometry, 2D graphing	Multivariable calculus, 3D geometry

Why It Matters

Quadrants appear constantly in algebra and trigonometry. In algebra, knowing the quadrant tells you the signs of coordinates, which is essential for graphing equations and inequalities. In trigonometry, the quadrant of an angle determines which trigonometric functions are positive or negative — a fact you rely on when solving equations, evaluating inverse trig functions, and working with the unit circle.

Common Mistakes

Mistake: Numbering the quadrants clockwise instead of counterclockwise.

Correction: Always start at the upper-right (Quadrant I) and move counterclockwise: I → II → III → IV. This matches the convention for measuring positive angles.

Mistake: Placing a point on an axis into a quadrant (e.g., saying (0, 4) is in Quadrant I).

Correction: Points on the x-axis or y-axis are not in any quadrant. Both coordinates must be nonzero for the point to belong to a specific quadrant.

Related Terms

x-y Plane — The plane that the four quadrants divide
Axes — The x- and y-axes that create the quadrants
Octants — The 3D equivalent with eight regions
Origin — The point (0,0) where the axes intersect
Coordinate — The x- and y-values that determine quadrant
Ordered Pair — The (x, y) notation used to identify points
Unit Circle — Uses quadrants to determine trig sign rules