Geometry Problem Solver What are the quadrants on a graph?
The two axes split the coordinate plane into four regions called quadrants. They are numbered I to IV starting at the top right and going anticlockwise. Each quadrant is defined by the signs of its coordinates: quadrant I is (+, +), II is (−, +), III is (−, −) and IV is (+, −).
Reading a point into its quadrant
A point is written (x, y), x first. Look at the two signs and the quadrant follows with no drawing required. The point (−4, 7) has a negative x and a positive y, so it is left of the vertical axis and above the horizontal one: quadrant II.
The order matters. (3, −5) and (−5, 3) sit in different quadrants, IV and II, even though they use the same two numbers.
Points on the axes belong to no quadrant
If either coordinate is zero, the point lies on an axis rather than inside a region. (0, 6) is on the y-axis, (−2, 0) is on the x-axis, and the origin (0, 0) is where the two meet. Exam questions like this one because the expected answer is that the point is in no quadrant at all.
Why anticlockwise
The convention comes from measuring angles, which also run anticlockwise from the positive x-axis. Quadrant I covers 0° to 90°, II covers 90° to 180°, III covers 180° to 270° and IV covers 270° to 360°. Once trigonometry starts, the same numbering tells you which functions are positive where, so it is worth getting the direction into memory now.
The four quadrants
| Quadrant | x | y | Example point | Angle range |
|---|---|---|---|---|
| I | Positive | Positive | (3, 4) | 0° to 90° |
| II | Negative | Positive | (−3, 4) | 90° to 180° |
| III | Negative | Negative | (−3, −4) | 180° to 270° |
| IV | Positive | Negative | (3, −4) | 270° to 360° |
Questions
Which quadrant is (0, 0) in?
None. The origin sits on both axes at once, and the quadrants are the open regions between the axes, not the axes themselves.
Why is quadrant II on the left instead of the right?
Because the numbering runs anticlockwise from the top right, so it moves I, II, III, IV as you go around. Reading them clockwise is the most common mistake in this topic.
Which quadrants are sine and cosine positive in?
Sine is positive in quadrants I and II, cosine is positive in I and IV, and tangent is positive in I and III. All three are positive only in quadrant I.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
