Rectangular coordinates say how far across and how far up. Polar coordinates say how far away and in what direction.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The notation
A polar point is (r, θ): r is the distance from the origin and θ is the angle from the positive x-axis.
Polar to rectangular
x = r cos θ and y = r sin θ. These are the unit circle coordinates, scaled up by r.
Rectangular to polar
r = √(x² + y²) and θ = arctan(y ÷ x), adjusted for the quadrant the point actually lies in.
Names are not unique
(3, 30°) and (3, 390°) are the same point. So is (−3, 210°), since a negative r points backwards.
When polar is better
Anything built around a centre — radar, a spiral, planetary orbits — is far simpler in polar form.
Distance and direction instead of across and up
A polar coordinate (r, θ) gives distance from the origin and angle from the positive x-axis. It is the natural system whenever a problem has rotational symmetry, which Cartesian coordinates handle awkwardly.
Converting between systems
x = r cos θ and y = r sin θ go one way; r = √(x² + y²) and θ = tan⁻¹(y/x) go the other. The inverse tangent needs adjusting for quadrant, since it only returns values in two of the four.
Polar coordinates are not unique
The same point has infinitely many representations: (2, 30°), (2, 390°), and (−2, 210°) are all the same place. Unlike Cartesian coordinates, polar ones do not name a point uniquely, which matters when solving equations.
Negative r is allowed
A negative r means going backwards along the ray, landing opposite the origin. It is a convention that keeps many polar graphs continuous, and ignoring it loses parts of some curves.
Step 2: Try It Yourself
Tap and try it out.
- Angle60° = π/3 rad
- x-coordinate0.500
- y-coordinate0.866
- cos 60°0.500
- sin 60°0.866
The cosine is the x-coordinate and the sine is the y-coordinate. They are not merely equal — they are the same two numbers, which is why angles past 90° cause no trouble.
Step 3: Watch an Example
One step at a time.
Watch Priya Convert (4, 30°)
Priya needs this polar point in rectangular form.
- Step 1
The formula for x is r cos θ, so x = 4 cos 30°.
Step 4: Your Turn
Practice makes it stick.
The Distance
Problem 1 of 2
The rectangular point (3, 4). What is r?
The Axis
Problem 2 of 2
The polar point (6, 0°). What is its x-coordinate?
Distance and Direction
1 of 8
(5, 90°) in polar. What is the y-coordinate?
2 of 8
(5, 90°) in polar. What is the x-coordinate?
3 of 8
The rectangular point (6, 8). What is r?
4 of 8
(2, 180°) in polar. What is the x-coordinate?
5 of 8
The rectangular point (0, 7). What is θ, in degrees?
6 of 8
The rectangular point (−5, 0). What is r?
7 of 8
Which polar pairs name the same point as (3, 30°)?
8 of 8
(4, 270°) in polar. What is the y-coordinate?
Step 5: Quick Check
Show what you know.
Question 1 of 2
The rectangular point (8, 6). What is r?
Question 2 of 2
What does r represent in a polar coordinate?
What You Learned
- A polar point is (r, θ): distance and direction.
- x = r cos θ and y = r sin θ convert to rectangular form.
- The same point has infinitely many polar names.