r = 5 is every point 5 from the origin, a circle. In rectangular form that same curve needs x² + y² = 25.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Lines through the origin
θ = 45° fixes the direction and leaves the distance free, drawing a line through the origin.
Rose curves
r = a cos(nθ) draws a rose. An odd n gives n petals; an even n gives 2n.
Cardioids and limaçons
r = a + b cos θ gives a limaçon, and when a equals b it is a heart-shaped cardioid.
Spirals
r = θ makes the distance grow with the angle, producing a spiral that never closes.
Negative r values
When the equation gives a negative r, the point is plotted in the opposite direction. That is how the extra petals appear.
The standard families
Circles, cardioids, limaçons and rose curves. A rose r = a cos(nθ) has n petals for odd n and 2n for even n, which is worth verifying by plotting rather than accepting.
Plot by table of angles
Choose angles, compute r, plot. One full revolution usually reveals the whole curve, though some need two. Sketching a few by hand builds intuition that recognising shapes cannot.
Symmetry tests save work
Replacing θ with −θ tests symmetry about the polar axis; replacing r with −r tests symmetry about the pole. Establishing symmetry first halves or quarters the plotting.
Simple only in polar form
r = θ is a spiral, trivially described and hopeless in Cartesian form. Choosing the coordinate system to suit the curve is the whole reason for having more than one.
Step 2: Try It Yourself
Tap and try it out.
- Angle0° = 0 rad
- x-coordinate1
- y-coordinate0
- cos 0°1
- sin 0°0
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 Diego Identify a Rose
Diego is given r = 4 cos(3θ).
- Step 1
The form a cos(nθ) identifies it as a rose curve.
Step 4: Your Turn
Practice makes it stick.
The Petals
Problem 1 of 2
r = 2 cos(4θ). How many petals?
The Circle
Problem 2 of 2
r = 7. What is the radius of this circle?
Name the Curve
1 of 8
r = 3 cos(5θ). How many petals?
2 of 8
r = 3 cos(2θ). How many petals?
3 of 8
r = 9. What is the radius?
4 of 8
r = 5 cos(7θ). How many petals?
5 of 8
r = 4 cos(6θ). How many petals?
6 of 8
r = 3 + 3 cos θ. Is it a cardioid? 1 yes, 0 no.
7 of 8
Match each polar equation with its curve.
Tap a card on the left to start.
8 of 8
r = 4 cos(3θ). What is r at θ = 0°?
Step 5: Quick Check
Show what you know.
Question 1 of 2
r = 5 cos(3θ). How many petals?
Question 2 of 2
What curve does r = 8 draw?
What You Learned
- r = a constant is a circle; θ = a constant is a line through the origin.
- r = a cos(nθ) is a rose: n petals for odd n, 2n for even n.
- A negative r plots in the opposite direction, which is how extra petals arise.