r = 5 is every point 5 from the origin, which is 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° is every point in one fixed direction, which draws a straight line through the origin.
Rose curves
r = a cos(nθ) draws a rose. If n is odd there are n petals; if n is even there are 2n.
Cardioids and limaçons
r = a + b cos θ gives a limaçon. When a and b are equal it is a cardioid, shaped like a heart.
Spirals
r = θ makes the distance grow with the angle, which produces a spiral that never closes.
How to sketch one
Build a table of θ against r, plot the points in order, and let the shape appear.
Curves that are hard in x and y
r = 3 is a circle, and r = θ is a spiral. Both are cumbersome in Cartesian form and trivial in polar. Choosing the coordinate system to suit the curve is the point of having more than one.
The standard families
Circles, cardioids, limaçons, and rose curves. A rose r = a cos(nθ) has n petals when n is odd and 2n when even, which is a striking fact worth verifying by plotting.
Plot by table
Choose angles, compute r, plot the points. Working through one full revolution usually reveals the whole curve, though some need two. Sketching a few by hand builds the intuition that pattern-matching alone cannot.
Test for symmetry first
Replacing θ with −θ tests symmetry about the x-axis; replacing r with −r tests symmetry about the origin. Establishing symmetry halves or quarters the plotting, which is worth the minute it takes.
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 Marcus Sketch r = 4 cos(3θ)
Marcus needs the shape without plotting dozens of points.
- Step 1
The form a cos(nθ) tells him this is a rose curve.
Step 4: Your Turn
Practice makes it stick.
The Circle
Problem 1 of 2
r = 7. What is the radius of this circle?
The Petals
Problem 2 of 2
r = 2 cos(4θ). How many petals does this rose have?
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? Enter 1 for yes, 0 for 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 when 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.
- r = a + b cos θ is a limaçon, and a cardioid when a equals b.