Polar Coordinate Grapher

About the Polar Coordinate Grapher

Plot polar curves r = f(θ) such as roses, cardioids and spirals (θ is entered as x).

How to use this calculator

1. Enter your polar equation using \(x\) for the angle, since this tool treats \(x\) as \(\theta\). For example, type \(2\cos(x)\) or \(1+\sin(x)\).
2. Check that the formula is written as a radius \(r\) in terms of the angle, not as \(y\) in terms of \(x\).
3. Graph the curve to see its shape, symmetry, and key features such as loops, petals, or spirals.
4. Use the displayed formula and example to compare your input with a known polar curve and confirm it matches what you expect.

The formula explained

A polar equation \(r=f(\theta)\) gives the distance \(r\) from the origin for each angle \(\theta\). The grapher converts those values into points and plots the full curve.

• r = the distance from the origin to the point
• \(\theta\) = the angle measured from the positive x-axis
• x = the input used for \(\theta\) in this tool
• \(f(\theta)\) = the rule that tells you the radius for each angle

Step by step method

1. Start with the polar equation and identify the radius formula \(r=f(\theta)\).
2. Choose several angle values, such as \(0\), \(\pi/2\), \(\pi\), and \(3\pi/2\), and compute the matching radii.
3. Plot each point using the angle and radius, then connect the points smoothly to reveal the curve.

Worked example

Problem. Graph the polar curve \(r=2\cos(x)\), where \(x\) represents \(\theta\).

1. At \(x=0\), \(r=2\cos(0)=2\), so the point is 2 units from the origin along the positive x-axis.
2. At \(x=\pi/2\), \(r=2\cos(\pi/2)=0\), so the curve passes through the origin.
3. These points form a circle centered at \((1,0)\) with radius 1, so the graph is a circle.
4. Answer: \(r=2\cos(x)\) graphs as a circle centered at \((1,0)\) with radius 1.

Tips and common mistakes

• In this tool, remember that \(x\) means \(\theta\), not the usual Cartesian x-coordinate.
• If your graph looks shifted or flipped, check for negative radii, because they can make points appear on the opposite side of the origin.

Frequently asked questions