Calculator
Example Data Table
| # | r1 | θ1 (deg) | r2 | θ2 (deg) | Distance (approx.) |
|---|---|---|---|---|---|
| 1 | 5 | 45 | 8 | 120 | 8.576 |
| 2 | 10 | 0 | 10 | 90 | 14.142 |
| 3 | 6 | 210 | 3 | 30 | 8.185 |
Values are rounded; your chosen precision may differ.
Formula Used
This calculator converts both polar points into Cartesian coordinates:
- x = r · cos(θ)
- y = r · sin(θ)
Then it applies the Euclidean distance formula:
d = √((x2 − x1)² + (y2 − y1)²)
For verification, it also computes the polar cosine rule:
d² = r1² + r2² − 2·r1·r2·cos(θ2 − θ1)
How to Use This Calculator
- Enter r1 and θ1 for the first polar point.
- Enter r2 and θ2 for the second polar point.
- Select Degrees or Radians for your angles.
- Choose a decimal precision level for rounding.
- Enable Normalize angles if you want standard θ values.
- Click Calculate Distance to view results above the form.
- Use Download CSV or Download PDF after calculation.
Article
Distance in Polar Coordinate Systems
Polar coordinates describe a point by radius r and angle θ. When you have two points, the distance is the straight-line separation in the plane, not the difference in radii. This calculator computes that separation while keeping your inputs in polar form.
Because angles repeat every full turn, θ and θ + 360° point the same way. With normalization enabled, the display stays readable while the math stays identical, which is helpful when copying values from logs or instruments quickly.
Why Convert to Cartesian Coordinates
The most reliable approach is converting each point to (x, y). Using x = r·cos(θ) and y = r·sin(θ), you get values that work with the familiar distance rule. This also makes debugging easy because you can see intermediate coordinates.
Cosine Rule Shortcut in Polar Form
A second method uses the angle difference Δθ = θ2 − θ1. Then d² = r1² + r2² − 2·r1·r2·cos(Δθ). The tool shows a cross-check distance from this formula, confirming the Cartesian result and helping you spot entry mistakes.
Angle Units and Normalization
You can enter angles in degrees or radians. Internally, calculations use radians, so degree inputs are converted automatically. If you enable normalization, angles are wrapped into a standard range, which keeps displays tidy for values like 450° or −30°. Normalization does not change the geometric direction.
Handling Negative Radius Values
Some datasets allow negative r. A negative radius represents the same point as a positive radius with the angle shifted by π. This calculator applies that convention, so distance stays correct even if one or both radii are negative.
Precision, Rounding, and Error Sources
Precision controls the number of decimals shown in results, including dx, dy, and distance. Higher precision is useful for small angle differences or large radii. Differences between the two distance methods are usually rounding, not a real mismatch.
Practical Uses in Engineering and Graphics
Polar distance appears in robotics, surveying, antenna patterns, navigation, and computer graphics. It helps compare two radial measurements taken from a common origin. After you compute a result, exporting to CSV or PDF supports reporting, lab notes, and repeatable calculations across projects.
FAQs
1) Can I enter one angle in degrees and the other in radians?
No. Select one angle unit and enter both angles in that unit. If your source data is mixed, convert one angle before calculating to keep results consistent.
2) Why does the calculator show two distance values?
The main distance comes from converting to Cartesian coordinates. The cross-check uses the polar cosine rule. They should match closely; small differences can occur from rounding at your chosen precision.
3) Does angle normalization change the computed distance?
No. Normalization only changes how the angles are displayed by wrapping them into a standard range. The direction is the same, so the final distance remains unchanged.
4) What happens if I enter a negative radius?
The tool converts negative radius values using an equivalent representation: r = |r| and θ = θ + π. This keeps the point location and the distance correct.
5) What if both points have the same angle?
If θ1 equals θ2, both points lie on the same ray from the origin. The distance becomes the absolute difference of the radii, because the points are collinear along that direction.
6) How do the CSV and PDF downloads work?
After you calculate, the result is stored in your session. The download buttons export the saved values as a simple CSV table or a one-page PDF summary for quick sharing and archiving.