Polar coordinates calculator

A polar coordinates calculator converts a point between polar form (r, θ), a distance and an angle, and rectangular (Cartesian) form (x, y). Our calculator converts both ways with exact answers such as (2√3, 2), in radians or degrees, and plots polar curves such as cardioids and roses, naming the curve and finding the area it sweeps out.

Choose To (x, y), To (r, θ) or Curve at the top. It opens on the rose r = 2 cos 3θ.

Try it

Polar Coordinates Calculator

Convert between polar and Cartesian coordinates and plot them

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How to use the polar coordinates calculator

1. Choose the mode: To (x, y) to convert polar to rectangular, To (r, θ) to convert rectangular to polar, or Curve to plot r as a function of θ.
2. Choose Radians or Degrees for the angle.
3. Type the values: r and θ, or x and y. Exact values such as π/6 and √3 work.
4. For a curve, type r in terms of θ, such as 1+cos(θ), and set how far round to draw, from 0 to anything between π/4 and 12π.
5. Read the answer, the working and the graph on a polar grid.

At multiples of π/6 and π/4 the answers are exact surds, with decimals alongside. For practice, Practice and help sets conversion questions with worked solutions.

The conversion formulas

r is the distance from the origin, and θ is the angle anticlockwise from the positive x-axis. When finding θ, check which quadrant the point is in: tan⁻¹ alone only gives angles between −90° and 90°.

Worked examples you can check in the calculator

Example 1: polar to rectangular

Convert (4, π/6) to rectangular coordinates.

The point is (2√3, 2), about (3.46, 2). In To (x, y) mode, type r = 4 and θ = pi/6 in radians.

Example 2: using degrees

Convert (6, 135°). Switch to Degrees and type 135. x = 6 cos 135° = −3√2 and y = 6 sin 135° = 3√2, so the point is (−3√2, 3√2).

Example 3: rectangular to polar, second quadrant

Convert (−3, 3) to polar coordinates. r = √(9 + 9) = 3√2. tan⁻¹(3/−3) = −45°, but the point is up and to the left, in the second quadrant, so θ = 135°, or 3π/4:

The calculator says which quadrant the point is in and gives the angle correctly.

Example 4: a negative angle

Convert (1, −√3) in degrees. r = 2 and θ = −60°. The calculator also gives 300°, the same direction measured from 0 to a full turn, since some courses want θ between 0° and 360°.

Check you have the method:

Plotting polar curves

In Curve mode, type r as a function of θ and the calculator plots it, names common shapes and says whether they are symmetric. Draw θ from 0 to 2π for one full turn.

The area comes from A = ½∫ r² dθ, adding thin sectors. Where r is negative, the point is plotted on the opposite side of the origin, which is how roses get their extra petals. The calculator works these areas out numerically for the range you draw.

Common mistakes

• Wrong quadrant for θ. Sketch the point. tan⁻¹ gives the wrong angle for points with negative x.
• Mixing radians and degrees. Set the calculator to match your question.
• Forgetting the square root for r. r = √(x² + y²), not x² + y².
• Swapping sin and cos. x uses cos and y uses sin.
• Drawing too little of a curve. Some curves need more than one turn to close, and roses can need 2π.

Practise in a short timed paper

Polar coordinates come up in our A-level Further Maths daily papers, and in College Calculus II & III in the US. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. For curves in x and y, use the Custom Graph calculator.

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