| Label | r | θ (degrees) | x = r·cos(θ) | y = r·sin(θ) |
|---|---|---|---|---|
| A | 5 | 30 | 4.3301 | 2.5000 |
| B | 6 | 120 | -3.0000 | 5.1962 |
| C | 4 | 210 | -3.4641 | -2.0000 |
| D | 7 | 315 | 4.9497 | -4.9497 |
These values assume degrees. Switch to radians when your angles come from calculus or physics problems.
Polar → Cartesian
Convert each polar point (r, θ) into Cartesian coordinates (x, y):
- x = r · cos(θ)
- y = r · sin(θ)
Degrees ↔ Radians
If angles are entered in degrees, convert before using trig:
- θ(rad) = θ(°) · π / 180
- θ(°) = θ(rad) · 180 / π
When “Connect points” is enabled, the calculator also estimates total path length using the distance formula on successive Cartesian points.
- Choose your angle unit: degrees or radians.
- Enter one or more points using r and θ.
- Enable optional plotting features like grid, axes, and connected paths.
- Press Plot and Convert to see the chart and table.
- Use Download CSV or Download PDF for reporting.
1) Understanding Polar Input
Polar coordinates describe a point using radius r and angle θ. This calculator accepts positive or negative r, so you can model reflections through the origin without rewriting angles. Each row is treated as an independent point with an optional label for plotting. You can enter up to 200 points in one run for dense shapes.
2) Choosing Degrees or Radians
Select degrees for common geometry tasks and radians for calculus, physics, and engineering formulas. Internally, angles are converted to radians before trigonometry using θ(rad)=θ(°)·π/180. The result table shows both degree and radian versions for quick verification.
3) Cartesian Conversion Output
For every input, the tool computes x=r·cos(θ) and y=r·sin(θ). Values are rounded to 10 decimal places to keep exports readable while staying accurate for most plotting and measurement work. This makes it easy to reuse coordinates in graphs, CAD sketches, or spreadsheets. Non‑numeric rows are flagged so you can fix data quickly.
4) Plot Scaling and Grid Reading
The canvas auto-scales to your points and adds padding so labels do not collide with the border. You can set the canvas size from 420–1600 px wide and 320–1200 px high. Optional grid and axes help you estimate slopes, quadrants, and symmetry at a glance, even when points span very different magnitudes.
5) Connecting Points and Measuring Paths
Enable “Connect points” to draw a polyline through your points in the entered order. The calculator then estimates total path length using the distance formula between consecutive Cartesian points. This is useful for tracing motion, plotting a route, or approximating piecewise curves.
6) Closing a Loop and Estimating Area
When “Close loop” is enabled and you have at least three points, the first and last points are joined to form a polygon. The tool reports a closed perimeter and polygon area using the shoelace method, giving fast feedback for shapes defined in polar form.
7) Exporting Results for Reports
After you calculate once, you can download CSV for spreadsheets and a compact PDF report for sharing. Exports include labels, r, input θ, converted x/y, and summary metrics such as min/max values and centroid. This keeps documentation consistent across projects.
1) What does a negative r mean?
A negative radius flips the point to the opposite direction on the same line. The calculator uses the given r directly in x=r·cos(θ) and y=r·sin(θ), so the plotted point lands correctly.
2) Should I enter θ in degrees or radians?
Use degrees for typical coordinate-geometry questions. Use radians when angles come from calculus, physics, or trig identities. Select the unit first, then enter θ values consistently for all rows.
3) Why do my points look “squished” on the plot?
The plot keeps a square aspect ratio to avoid distortion. If one axis range is much larger, points can appear clustered. Increase the canvas size or enable the grid to read positions more clearly.
4) How is path length calculated?
When “Connect points” is on, the tool converts each polar point to Cartesian, then sums distances between consecutive points using √((Δx)²+(Δy)²). It is an estimate based on straight segments.
5) How is polygon area computed when I close the loop?
With three or more points, the first and last points are joined and the area is computed from Cartesian coordinates using the shoelace method. The reported area is the absolute value, so orientation does not change it.
6) What do the CSV and PDF exports include?
Exports include labels, r, θ input, θ in degrees and radians, x, y, and summary metrics like min/max and centroid. Downloads always use the most recent calculation stored in your session.