• 51 ideas
  • Checked October 2026

Polar Graph Art Project Ideas

Press for a picture to plot on polar graph paper, with the equations that draw it.

Polar Graph Art Project Ideas

More filters
Show all 51
  1. A clock mainspring coiled in its drumSpring r = 0.4θ from 0 to 8π, drum r = 10.5, and the small circle r = 0.5 for the pin in the middle.

Next step Pythagorean spiral art ideasMore math drawing ideas

All 51 ideas

Flowers from rose curves 13

  • A daisy-rimmed dial reading three o'clock Petals r = 6 cos(6θ), hub r = 1, a long hand along θ = 90° to r = 5, a short hand along θ = 0° to r = 3.
  • A flower with two layers of petals r = 5 cos(4θ) and r = 3 sin(4θ). The sine petals are shorter and sit in the gaps between the others.
  • A lucky clover: a leaf in each quarter and a stalk r = 4 sin(2θ) puts one leaf in each quarter. The stalk is the ray θ = 270°, from the middle out to r = 7.
  • A pond flower inside three widening rings Petals r = 3 cos(4θ) and r = 2 sin(4θ). The ripples are the circles r = 4, r = 5.5 and r = 7.
  • A six-petal blossom whose petals stop short of the middle r = 3 + 2 cos(6θ). A circle of radius 3 with six bumps. The 6 sets how many, the 2 how deep.
  • A twelve-petal daisy with a shaded disc r = 6 cos(6θ) for the petals and the small circle r = 1.5 for the center. Shade the circle.
  • A wind rose with eight petals inside a double ring r = 6 cos(2θ) points north, east, south and west; r = 3 sin(2θ) fills the diagonals; rings at r = 6.5 and r = 7.
  • An eight-petal daisy r = 5 cos(4θ). Four is even, so there are eight petals. Every petal tip touches the circle r = 5.
  • Eight petals fitted inside a ring, tip to rim r = 5 cos(4θ) inside the circle r = 5. The rose fits its circle exactly: every tip meets the rim.
  • Five petals from a sine rose, one aimed straight up r = 4 sin(5θ). Sine turns the flower so that one petal points straight up. Try cos to see it turn.
  • The quadrifolium: four petals in a cross r = 4 cos(2θ). An even number beside θ gives twice as many petals: 2 makes four.
  • Six overlapping petals from a rose with a fraction r = 4 cos(3θ/2), with θ from 0° to 720°. A fraction beside θ makes petals that cross each other.
  • The trifolium: three petals from one cosine r = 4 cos(3θ). An odd number beside θ gives that many petals. Change the 4 to make them longer.

Hearts, apples and shells 12

  • A cardioid apple with a leaf at the top Apple r = 3 - 3 sin θ. Stalk: the ray θ = 90° out to r = 2. Leaf: r = 2 sin(2θ), drawn only from 0° to 90°.
  • A floating leaf with one notch, from a cardioid r = 3 + 3 cos θ. The same cardioid turned on its side, so the notch opens to the left.
  • A gumdrop with a flat base r = 4 + 2 sin θ. Once the first number is double the second, the dimple of the limaçon has flattened out.
  • A heart inside a heart r = 4 - 4 sin θ and r = 2 - 2 sin θ. Both dents meet at the middle of the paper.
  • A kidney bean with a dent in one side r = 3 + 2 cos θ. A limaçon whose first number is a little bigger than its second has a dimple, not a point.
  • A limaçon knotted into an inner loop r = 2 + 4 cos θ. When the second number is the bigger one, the limaçon grows an inner loop.
  • A party balloon from a limaçon, string straight down Balloon r = 5 + 2 sin θ, a limaçon with no dimple. The string is the ray θ = 270°, drawn from r = 3 out to r = 9.
  • A plump heart from one cardioid r = 3 - 3 sin θ. One dent, at the top, and a round base. The name cardioid comes from its heart-like shape.
  • A shell of three nested loops r = 1 + 2 cos θ, r = 2 + 4 cos θ and r = 3 + 6 cos θ: the first one doubled, then tripled, inner loop and all.
  • A sliced apple with its core and two pips Skin r = 5 - 5 sin θ, core r = 2 - 2 sin θ, stalk θ = 90° to r = 2, pips: the petals of r = 3 cos(6θ) at 240° and 300°.
  • An owl mask: dimpled oval, ring eyes, small beak Face r = 5 - 3 sin θ, eyes r = 2 cos θ and r = -2 cos θ, beak r = 1.5 sin(3θ) drawn from 240° to 300°.
  • Four hearts overlapping in a ring r = 3 - 3 sin θ, r = 3 + 3 sin θ, r = 3 - 3 cos θ and r = 3 + 3 cos θ. All four dents meet in the middle.

Spirals and snails 13

  • A clock mainspring coiled in its drum Spring r = 0.4θ from 0 to 8π, drum r = 10.5, and the small circle r = 0.5 for the pin in the middle.
  • A coiled rope from the Archimedean spiral r = 0.5θ, with θ in radians from 0 to 6π. The coils are evenly spaced; a bigger number spreads them out.
  • A lollipop on a stick Swirl r = 0.5θ from 0 to 6π, rim r = 9.5, and for the stick the ray θ = 270° from r = 9.5 to r = 16.
  • A nautilus shell from a logarithmic spiral r = e^(0.2θ), with θ in radians from 0 to 4π. Each turn is wider than the one before it.
  • A shepherd's crook from the lituus r = 4/√θ, with θ in radians from 0.1 to 4π. A long handle that curls in at one end.
  • A spider at the hub of a spiral web Spokes along every 30° line, web r = 0.4θ from 0 to 8π, body r = 1 and eight legs r = 2.5 cos(4θ).
  • A spiral galaxy with a bright core Arms r² = 4θ from 0 to 4π, which is r = 2√θ and r = -2√θ, and the small circle r = 1 for the core.
  • A sunflower with seeds, petals and a stem Seeds r² = 2θ from 0 to 4π, ring r = 5.5, petals r = 7.5 + 2 cos(12θ), stem the ray θ = 270° from r = 9.5 out.
  • A whirlpool from the hyperbolic spiral r = 6/θ, with θ in radians from 0.5 to 6π. It starts far out and winds in toward the middle without reaching it.
  • An open bloom: a spiral ringed by three wavy edges Heart r = 0.3θ from 0 to 4π, then three wavy rings: r = 5 + cos(5θ), r = 7 + sin(5θ) and r = 9 + cos(5θ).
  • Fermat's spiral, a pair of arms from one point r² = 4θ, with θ in radians from 0 to 4π. Plot r = 2√θ and r = -2√θ: two arms that meet in the middle.
  • Snail-shell loops from the cochleoid r = 6 sin θ / θ, with θ in radians from 0.01 to 6π. Each loop is smaller than the last, and all pass through the middle.
  • Two nested spirals, one for each colored pencil r = 0.5θ and r = -0.5θ, both from 0 to 6π. The second is the first turned half a circle, so they nest.

Butterflies, bows and loops 13

  • A bow tie from the lemniscate r² = 16 cos(2θ). Plot only where cos(2θ) is positive. Swap cos for sin to tilt the bow 45°.
  • A butterfly from one long equation r = e^(sin θ) - 2 cos(4θ) + sin^5((2θ - π)/24). Run θ from 0 to 24π; the last term layers the wings.
  • A cartwheel with a spoke along every 30° line Rim r = 9 and r = 8, hub r = 1, and a spoke along every 30° line from the hub out to r = 8.
  • A cog wheel with twelve teeth r = 6 + 0.5 cos(12θ). A circle with twelve small waves. A bigger number than 0.5 makes longer teeth.
  • A dragonfly with two pairs of wings r² = 25 cos(2(θ - 15°)) and r² = 25 cos(2(θ + 15°)), with the line θ = 90° through the middle for the body.
  • A five-armed sea star with rounded tips r = 3 + 2 cos(5θ). The 5 counts the arms and the 2 sets how far they reach past the circle r = 3.
  • A grinning head wearing two circle lenses Head r = 10, lenses r = 4 cos θ and r = -4 cos θ, and for the smile the circle r = 6 drawn only from 230° to 310°.
  • A hair bow with two tails Bow r² = 16 cos(2θ). The tails are the rays θ = 250° and θ = 290°, each from the middle out to r = 5.
  • A pair of round glasses, seen from the front r = 4 cos θ and r = -4 cos θ. Each is a circle through the middle of the paper, so the two lenses touch there.
  • A peanut shell r = 4 + 2 cos(2θ). Widest along the 0° line and pinched to a waist at 90°. A bigger second number pinches harder.
  • A ribbon rosette with small loops inside big ones r = 1 + 3 cos(3θ). Three big loops, each with a small loop tucked inside it, because 3 is more than 1.
  • A snowman from two circles that touch r = 4 sin θ is the head, above the middle of the paper; r = -6 sin θ is the body, below it.
  • An atom sign: three lemniscates around a dot r² = 25 cos(2θ), r² = 25 cos(2(θ - 60°)) and r² = 25 cos(2(θ - 120°)), with the small circle r = 0.5 in the middle.

How it works

Fifty-one pictures to plot on polar graph paper, each with the equations that draw it: flowers from rose curves, hearts and shells from cardioids and limaçons, spirals, butterflies and bows. Every equation was plotted before it was listed.

Read the full guide

Change the numbers

Many hints say what one number in the equation does. In a rose curve such as r = 4 cos(3θ), an odd number beside θ gives that many petals and an even number twice as many. In a limaçon such as r = 3 + 2 cos θ, the two numbers decide between a dimple, a point and an inner loop.

From one equation to a whole scene

Level, under More filters, sorts by how many equations a picture needs. A butterfly from one long equation is a single curve; a spider at the hub of a spiral web takes spokes, a spiral, a circle and a rose. The list gives equations, not a printable sheet.

  • The spirals use θ in radians: at θ = 2π the coiled rope has gone around once.

Questions and answers

What can you draw with polar equations?

Flowers, hearts, shells, spirals, butterflies and bows. A rose curve draws petals, a cardioid draws a heart or an apple, a limaçon draws a bean or a loop inside a loop, and a lemniscate draws a bow tie.

How many petals does a rose curve have?

For r = a cos(kθ) with k a whole number, k petals when k is odd and 2k petals when k is even. So r = 4 cos(3θ) has three petals and r = 4 cos(2θ) has four.

What polar equation makes a heart?

The cardioid r = 3 - 3 sin θ: a heart with one dent at the top and a round base. The curve is named for its heart-like shape but is as close to an apple, so the list also plots it with a leaf.

What is the equation of the butterfly curve?

r = e^(sin θ) - 2 cos(4θ) + sin^5((2θ - π)/24), with θ running from 0 to 24π. It dates from 1989. The last term is there for looks: it shifts each lap a little, so the wings come out in layers.

More tools

Show 4 more

Browse Drawing Ideas