Rectangular coordinates say how far across and 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), with r the distance from the origin and θ 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 by r.
Rectangular to polar
r = √(x² + y²) and θ = arctan(y ÷ x), adjusted for the quadrant the point is actually in.
Names are not unique
(3, 30°) and (3, 390°) name the same point, and so does (−3, 210°). A negative r points backwards.
When polar helps
Anything organised around a centre is simpler in polar form. A circle becomes r = 5 rather than x² + y² = 25.
Distance and direction
A polar coordinate (r, θ) gives distance from the origin and angle from the positive x-axis. It is the natural system whenever a situation has rotational symmetry, which Cartesian coordinates handle clumsily.
Converting
x = r cos θ and y = r sin θ go one way; r = √(x² + y²) and θ = tan⁻¹(y/x) return, with the angle adjusted for quadrant since the inverse tangent covers only two of the four.
Representations are not unique
The same point can be written with different angles or a negative r. Unlike Cartesian coordinates, polar ones do not name a point uniquely, which matters when solving polar equations.
Negative r means backwards
A negative r places the point opposite the direction θ indicates. The convention keeps many polar curves continuous, and ignoring it loses whole petals of some rose 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 Sana Convert a Polar Point
Sana converts (4, 30°) to 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 Coordinate
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
(4, 270°) in polar. What is the y-coordinate?
7 of 8
Which polar pairs name the same point as (3, 30°)?
8 of 8
The rectangular point (−5, 0). What is r?
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.
- Every point has infinitely many polar names.