Quadrants of Coordinate Plane - I, II, III, IV

Geometry

TL;DR
The coordinate plane is divided by the x-axis and y-axis into four quadrants, numbered I, II, III, IV counterclockwise starting from the upper right. Each quadrant has a specific sign convention for (x,y): Quadrant I: both positive; II: x negative, y positive; III: both negative; IV: x positive, y negative.

What Is a Quadrant?

A quadrant is one of the four regions into which the coordinate plane is divided by the two coordinate axes — the horizontal x-axis and the vertical y-axis. The word quadrant comes from the Latin quadrans, meaning "a quarter" — and each quadrant is exactly one-quarter of the plane.

When the two axes intersect at right angles, they cut the plane into four equal infinite regions. Each region is labelled with a Roman numeral — I, II, III, IV — following a strict counterclockwise convention that traces back to René Descartes's 1637 La Géométrie.

A point in the coordinate plane is written as an ordered pair (x,y), where x is the horizontal distance from the y-axis and y is the vertical distance from the x-axis. The signs of x and y — positive or negative — determine which quadrant the point lies in.

Four Quadrants in Coordinate Plane

The four quadrants are arranged counterclockwise around the origin:

• Quadrant I (upper right): the region where x > 0 and y > 0. Every point here has both coordinates positive. Example: (3,5).
• Quadrant II (upper left): the region where x < 0 and y > 0. Negative x, positive y. Example: (−3,5).
• Quadrant III (lower left): the region where x < 0 and y < 0. Both coordinates negative. Example: (−3,−5).
• Quadrant IV (lower right): the region where x > 0 and y < 0. Positive x, negative y. Example: (3,−5).

The counterclockwise convention is not arbitrary — it matches the standard direction of positive angle measurement in trigonometry.

Sign Convention in Quadrants

The sign of the x-coordinate and the y-coordinate uniquely determines a point's quadrant.

Pattern: starting from Q I with both signs positive, each subsequent quadrant flips one sign as you move counterclockwise: (+,+)→(−,+)→(−,−)→(+,−).

What Is Origin?

The origin is the point (0,0) — the single point where the x-axis and y-axis intersect.

Properties of the origin:

• Its coordinates are both zero: (0,0).
• It does not belong to any of the four quadrants.
• It is the reference point from which every other point's coordinates are measured.

Plotting Points on Quadrants

To plot a point (x,y) in the coordinate plane:

Step 1. Start at the origin (0,0). Step 2. Read the x-coordinate and move horizontally — right if x is positive, left if x is negative. Step 3. Read the y-coordinate and move vertically — up if y is positive, down if y is negative. Step 4. Mark the point.

Trigonometric Values in Different Quadrants

The ASTC rule summarizes which trig functions are positive in each quadrant:

• Q I — All: all six trig functions are positive.
• Q II — Sine: sine is positive; the other four are negative.
• Q III — Tangent: tangent is positive; the other four are negative.
• Q IV — Cosine: cosine is positive; the other four are negative.

Why this pattern.

The signs of sin, cos, and tan follow the quadrant.

Common Mistakes With Quadrants

• Numbering quadrants clockwise instead of counterclockwise.
• Putting axis points in a quadrant.
• Confusing Q II with Q IV (or Q I with Q III).

Key Takeaways

• Four quadrants, numbered I, II, III, IV counterclockwise from the upper right.
• Sign conventions: Q I (+,+), Q II (−,+), Q III (−,−), Q IV (+,−).
• The origin (0,0) does not belong to any quadrant.
• Axis points belong to no quadrant — only to the x-axis or y-axis.