Toolarithm
Editable Polar Paper A3, 5mm, Blue, 36 Spokes
Editable A3 preset

A3 5mm Blue Polar Paper — 36 Spokes

Sample a cardioid using ten-degree references and keep its zero-radius cusp explicit. The broad A3 background supports the construction while you supply the calculated points.

Free to use
Print ready
PDF, PNG, SVG
Custom sizes

Polar Grid Preview

Choose your radial and angular divisions, then export an accurate plotting sheet.

Preparing the live paper preview…

What Can You Create?

A cardioid with a named formula

Write the chosen radial rule and its units before marking samples. A smooth-looking closed outline does not establish that it follows the formula. Keep a small value table beside the plot and distinguish those computed coordinates from the illustrative stroke joining them.

A cusp identified at the origin

Include the angle where the radius becomes zero instead of joining neighboring samples across it without explanation. The origin marker is a reference rather than an automatically constructed cusp. Label the relevant input angle and keep the reason that its point reaches the center in the worksheet.

Near-origin interpolation below the first ring

Some calculated radii can be smaller than one five-millimeter interval. Locate them inside the first circle rather than rounding them away. The ten-degree spokes provide direction, while the authored calculation identifies the small distance and any stated plotting precision.

Why Users Love This Tool

Fine angular sampling on regular blue references

Thirty-six initial spokes divide the turn into ten-degree intervals. Seven-point degree labels appear at each spoke and the origin marker is enabled. The #3B82F6 strokes are 0.35pt with five-millimeter ring spacing. Try a near-origin point label and adjust text frequency or line treatment if the annotations become crowded.

A3 field capacity stated around its starting labels

Original stock is 297×420 mm297 \times 420\,\mathrm{mm} with linked twelve-millimeter margins and center (148.5,210) mm\left(148.5, 210\right)\,\mathrm{mm}. Twenty-five circles reach a 125mm outer radius under the degree-label preset. The cardioid's maximum radius is eighty millimeters. Label, margin, paper, or stroke changes can alter fitting, so inspect current circle allowance before adopting the original count.

Perfect For

Learners tracing a curve through a zero-radius point

Retain the zero-radius input in the sample table even though its plotted point is the origin. Compare neighboring angles to see why the center belongs to the intended curve. The blank grid provides references but does not choose the curve's sample method or generate its cusp.

Teachers discussing finite sampling

Show the cardinal samples and a pair near 180 degrees, then ask which features an incomplete trace might miss. Keep approximations and point precision stated. A finite set of samples supports the illustration but does not make a fitted connecting line an exact curve automatically.

Authors retaining an annotated cardioid explanation

Keep formula, angle convention, coordinate marks, and connecting-stroke roles visible in the master. Reserve a note area for the near-origin calculation instead of forcing all labels into the crowded center. Save that composed explanation beside its selected background configuration.

Worked cardioid: r=40 (1+cos⁡ ⁣(θ))r = 40\,\left(1 + \cos\!\left(\theta\right)\right) millimeters

Take θ in degrees\theta\,\mathrm{in}\,\mathrm{degrees} for these samples. At 0∘0^{\circ}, 90∘90^{\circ}, 180∘180^{\circ}, and 270∘270^{\circ}, radii are 80, 40, 0, and 40mm, corresponding to sixteen rings, eight rings, the origin, and eight rings. Original A3 page positions are (228.5,210)\left(228.5, 210\right), (148.5,170)\left(148.5, 170\right), (148.5,210)\left(148.5, 210\right), and (148.5,250) mm\left(148.5, 250\right)\,\mathrm{mm}. At 60∘60^{\circ} the radius is 60mm and at 120∘120^{\circ} it is 20mm. Near the cusp, 170∘170^{\circ} and 190∘190^{\circ} both give 40 (1−cos⁡ ⁣(10∘))≈0.6077 mm40\,\left(1 - \cos\!\left(10^{\circ}\right)\right) \approx 0.6077\,\mathrm{mm}, well inside the first five-millimeter circle. The ideal curve reaches the origin at 180∘180^{\circ}, so include that sample explicitly when composing the trace. The radius stays between zero and eighty millimeters, within the original 125mm field. Add the formula, points, table, and connecting curve yourself; the generator exports only the selected circular reference.

Original preset specifications

This reference describes the named starting configuration. The live preview and downloads follow your current settings.

Original paper
A3 portrait · 297×420 mm297 \times 420\,\mathrm{mm}
Page insets
Top 12, right 12, bottom 12, left 12 mm
Inset rectangle
273×396 mm273 \times 396\,\mathrm{mm}
Radial field
25 rings · 5mm gaps · 125mm outer radius
Angular interval
36 spokes · 10∘10^{\circ} per sector
Original labels
degrees · 7pt · on; origin on
Polar stroke
#3B82F6 · 0.35pt
Original export sizes
PDF/SVG 297×420 mm297 \times 420\,\mathrm{mm}; PNG 3508×4961 px3508 \times 4961\,\mathrm{px} at nominal 300 DPI
How It Works

How it works in three quick steps.

1

Write the cardioid rule and required samples

State r=40 (1+cos⁡ ⁣(θ)) mmr = 40\,\left(1 + \cos\!\left(\theta\right)\right)\,\mathrm{mm} and the angle convention. Include the cardinal directions and 180∘180^{\circ} zero-radius point in the sample plan. Keep the maximum eighty-millimeter radius separate from the much smaller values chosen near the origin.

2

Mark the cusp and interpolate nearby radii

Locate the sixteen-ring maximum and eight-ring side samples, then place the 170- and 190∘190^{\circ} values inside the first circle. Label the cusp input independently from the background origin marker. Adjust guide and angular text treatments around the actual point annotations.

3

Proof the completed trace explanation

Export current A3 settings and compose the curve at the full page extent. Measure sixteen original radial intervals as eighty millimeters. Save the sample table, plotting precision, annotated trace, and selected background address together after checking the crowded center labels.

Download & Print

PDF for the cusp-sampling worksheet

The initial vector PDF preserves a 297×420 mm297 \times 420\,\mathrm{mm} A3 page with blue polar references. Add computed coordinates and the table without reducing that extent when keeping radial millimeters. Match stock and Actual Size, then inspect the origin annotation and longest radius on the final copy.

PNG beneath a cardioid study document

Original portrait dimensions are 3508×49613508 \times 4961 pixels at nominal 300 DPI. Keep the whole A3 image with its existing margins during placement. A receiving-app fit can alter small near-origin radii, so retain their written values and measure the completed plot after composition.

SVG for separate sample and trace layers

Use independent layers for blue references, computed points, cusp label, connecting stroke, and formula. Preserve physical A3 dimensions while refining the illustration. The composed master contains the mathematical construction that the blank vector export and background Share address cannot store.

Check the finished paper

  1. Preserve the selected A3 extent through curve composition and choose physical paper matching the resulting page.
  2. Use 100% or Actual Size and verify the origin-to-sixteenth-ring maximum as eighty millimeters.
  3. Read the 180∘180^{\circ} cusp statement and approximately 0.6077mm neighboring samples separately from the background origin marker.
  4. Keep the formula, degree convention, and stated plotting precision visible beside the final computed sample trace.

Adobe's PDF page-sizing guidance describes scaling controls. W3C's paper-size definitions document standard page dimensions.

FAQ

Frequently Asked Questions

Why is the 180∘180^{\circ} point called the origin cusp?
At 180 degrees, cosine is minus one, so forty times one plus cosine gives zero radius. The ideal cardioid therefore reaches the origin at that input, where its curve has a cusp. The printed origin marker only identifies the reference center; it does not generate or prove the curve feature. Include the zero-radius sample and formula in the annotated explanation instead of relying on the background marker to communicate it.
How should a radius smaller than the first circle be plotted?
It lies between the origin and the five-millimeter first ring. The 170- and 190∘190^{\circ} samples are approximately 0.6077mm from the center, not on the first circle and not exactly zero. Retain the written values and state the practical plotting precision used for the illustration. Printed rings provide reference intervals rather than quantizing every allowed coordinate, so deliberately distinguish a small positive radius from the cusp sample at the origin.
Can joining neighboring samples omit the intended cusp?
If the 180∘180^{\circ} origin sample is omitted and the near-origin points are simply connected, the illustrative line may bypass the feature that the formula specifies. Include the zero-radius input explicitly and retain the connecting method beside the table. Ten-degree references make a sample sequence convenient, but they do not evaluate the curve or guarantee that a manually drawn connection reproduces every feature of the continuous cardioid.
What limits the formula's radius range?
Cosine lies between minus one and one, so one plus cosine lies between zero and two. Multiplying by forty millimeters gives radii between zero and eighty millimeters. This mathematical range is independent of the original 125mm outer field, which only supplies paper allowance. No negative-radius reversal is needed for this stated formula. Keep the range derivation beside the plotting convention instead of inferring its maximum from the last circle printed.
Does the background link save the cardioid's formula and cusp?
Share stores the validated polar settings such as paper, spacing, spokes, labels, strokes, origin treatment, and margins. It does not store the formula, calculated samples, curve, or cusp annotation supplied outside the generator. Retain the completed mathematical master separately with its table and plotting precision. Opening the background address reconstructs the blue reference layer, while the annotated file explains which curve and sample method were intended.

About This Tool

A3 five-millimeter blue polar paper with thirty-six spokes provides a broad reference for a cardioid study that keeps its sampled points and origin feature explicit. Ten-degree directions help locate a convenient input sequence, while the five-millimeter rings establish physical radial intervals. The mathematical rule remains an authored statement beside the plot rather than something inferred from the circular background or a finished-looking connecting line.

The worked formula gives a maximum eighty-millimeter radius and a zero-radius point at 180 degrees. Its neighboring samples lie less than one millimeter from the origin, making the distinction between a small positive radius and the cusp important. Including that input explicitly prevents a simple connection of surrounding points from silently skipping the intended center feature. The cardinal page locations and written precision provide additional checks on how the illustration was composed.

A retained explanation needs the formula, sample table, cusp label, and connecting method alongside the reproducible blue field. Test the crowded center annotations, inspect the current layout after text or page edits, and preserve full A3 placement when radial millimeters matter. Measure the completed maximum run and retain any stated approximation precision. Saving that annotated curve with its background settings address preserves both repeatable references and the mathematical context needed to reconstruct the cardioid independently from the editor.

Original Preset Reference

Polar Paper A3, 5mm, Blue, 36 Spokes; 5mm circles, Blue Lines polar grid reference for worked cardioid: r = 40 (1 + cos (θ)) millimeters
Printable polar graph paper for A3 worksheets with 5mm circles, 36 spokes, and blue guides for trigonometry, physics, and scientific plotting.
Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Paper dimensions, margins, guide geometry, preview rendering, and exports are derived from the same typed settings and checked at Actual Size.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.