Toolarithm
Editable Polar Paper Letter, 10mm, Gray, 12 Spokes
Editable Letter preset

Letter 10mm Gray Polar Paper — 12 Spokes

Connect six equally spaced points on one circle. Compare the hexagon's straight perimeter with the longer circular circumference while retaining a clear radius label.

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Polar Grid Preview

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

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What Can You Create?

A regular hexagon from alternate spokes

Choose every second printed direction while retaining one radius for all vertices. The resulting six angular gaps are sixty degrees. Join consecutive points with straight chords and label the selected sequence, rather than allowing every one of the twelve background spokes to imply a polygon vertex.

Straight and curved boundary measurements

Compare the six chord lengths with the complete circle path. The same vertices and radius can support two different boundaries, so name which is being measured. A chord is not its corresponding arc even when their endpoints are shared and both are shown on the same gray reference.

Six equal triangles with a common center

Connect the origin to the selected vertices when explaining why a sixty-degree chord equals the radius. Those authored center lines subdivide the hexagon into a geometric argument. Keep the selected line convention clear so unused reference spokes are not counted as additional polygon sides or triangles.

Why Users Love This Tool

A coarse radial rhythm under the polygon outline

Original circles are ten millimeters apart, with #525252 strokes at 0.35pt. Twelve spokes give thirty-degree divisions, while the hexagon uses sixty-degree steps. Seven-point degree labels are shown at every initial direction and the origin marker is enabled. Try a darker chord outline and radius label against that reference treatment.

A Letter center and explicit original ring allowance

Starting stock is 215.9×279.4 mm215.9 \times 279.4\,\mathrm{mm} with linked twelve-millimeter borders and center (107.95,139.7) mm\left(107.95, 139.7\right)\,\mathrm{mm}. Eight circles reach an eighty-millimeter radius under the original label layout. The fourth-ring hexagon uses half that radial extent. Text, stroke, border, or paper edits can change fitting, so inspect the current background before retaining the written original circle count.

Perfect For

Students comparing polygon and circle boundaries

Name the selected radius, one straight side, and the two complete boundary totals. Explain why equal chord endpoints do not create equal arc and chord lengths. The regular reference supports the picture while the formula labels make its measurement distinctions explicit.

Teachers explaining sixty-degree geometry

Use the selected center angles to justify six equal triangles and the chord-to-radius relationship. Ask which of the twelve spokes become the six authored center connections. The blank polar pattern does not draw a hexagon or supply the student's boundary calculation.

Authors retaining a labeled comparison master

Keep the curved-circle reference distinguishable from the straight polygon outline in the final copy. Write the perimeter and circumference units beside their formulas. Save the complete diagram with its selected background address so the intended six-point construction survives beyond the blank reference export.

Worked hexagon: fourth-ring vertices at sixty-degree increments

Select radius 40mm, the fourth original circle, and mark 0∘0^{\circ}, 60∘60^{\circ}, 120∘120^{\circ}, 180∘180^{\circ}, 240∘240^{\circ}, and 300∘300^{\circ}. Their upward Cartesian offsets are (40,0)\left(40, 0\right), (20,20 3)\left(20, 20\,\sqrt{3}\right), (−20,20 3)\left(-20, 20\,\sqrt{3}\right), (−40,0)\left(-40, 0\right), (−20,−20 3)\left(-20, -20\,\sqrt{3}\right), and (20,−20 3) mm\left(20, -20\,\sqrt{3}\right)\,\mathrm{mm}. Consecutive vertices have a sixty-degree central separation, making each straight chord 2×40×sin⁡ ⁣(30∘)=40 mm2 \times 40 \times \sin\!\left(30^{\circ}\right) = 40\,\mathrm{mm}. The regular hexagon perimeter is therefore 6×40=240 mm6 \times 40 = 240\,\mathrm{mm}. The containing circle's circumference is 2 π×40=80 π≈251.327 mm2\,\pi \times 40 = 80\,\pi \approx 251.327\,\mathrm{mm}, approximately 11.327mm longer. The vertical vertex-offset magnitude is approximately 34.641mm, distinct from the radius and side length. Add the six vertices, chords, optional center connections, and formulas yourself. The original gray export contains the circular reference rather than an automatically constructed polygon.

Original preset specifications

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

Original paper
Letter portrait · 215.9×279.4 mm215.9 \times 279.4\,\mathrm{mm}
Page insets
Top 12, right 12, bottom 12, left 12 mm
Inset rectangle
191.9×255.4 mm191.9 \times 255.4\,\mathrm{mm}
Radial field
8 rings · 10mm gaps · 80mm outer radius
Angular interval
12 spokes · 30∘30^{\circ} per sector
Original labels
degrees · 7pt · on; origin on
Polar stroke
#525252 · 0.35pt
Original export sizes
PDF/SVG 215.9×279.4 mm215.9 \times 279.4\,\mathrm{mm}; PNG 2550×3300 px2550 \times 3300\,\mathrm{px} at nominal 300 DPI
How It Works

How it works in three quick steps.

1

Select a common radius and six directions

Write the forty-millimeter radius and choose every second original spoke. List the selected angles, including the return from three hundred degrees to zero. Keep the six vertex count separate from the twelve directions printed by the background.

2

Draw chords and label both boundary meanings

Connect consecutive vertices with straight segments and keep the circle visible as a separate path. Add a radius and one side label before computing perimeter and circumference. Try their outline styles against the gray references and edit the guide appearance when needed.

3

Proof the measured hexagon comparison

Export the current Letter sheet and compose the six-point study at full-page size. Measure the fourth-ring radius as forty millimeters on paper. Read the straight-versus-curved boundary labels and save the annotated comparison beside its chosen background address.

Download & Print

PDF for the polygon-and-circle worksheet

Original PDF stock is Letter at 215.9×279.4 mm215.9 \times 279.4\,\mathrm{mm} with vector gray circles, spokes, and labels. Keep that full extent when adding the hexagon. Match the paper and use Actual Size for ten-millimeter rings, then inspect where a chord and circle arc share endpoints.

PNG beneath a boundary-measurement exercise

The initial portrait image is 2550×33002550 \times 3300 pixels at nominal 300 DPI. Preserve its complete borders and Letter proportions during placement. A fitted worksheet image can shrink both boundaries, so measure the final composed radius after adding formulas and polygon labels.

SVG for separately styled straight sides

Keep the gray polar field under distinct hexagon sides, vertex markers, center connections, and measurement labels in a vector master. Maintain physical Letter dimensions. The annotated file stores the selected six points and boundary convention that the blank export and Share address do not contain.

Check the finished paper

  1. Preserve the selected Letter extent throughout composition and match the physical stock to the exported page.
  2. Choose 100% or Actual Size and check the origin-to-fourth-circle radius as forty millimeters on the completed copy.
  3. Read all six vertex directions and distinguish the straight 240mm perimeter from the approximately 251.327mm circumference.
  4. Keep chord, arc, radius, and complete boundary meanings labeled before retaining or duplicating the comparison worksheet.

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

FAQ

Frequently Asked Questions

Why does each regular-hexagon side equal the radius here?
Adjacent selected vertices are sixty degrees apart at the center. Their chord length is twice the forty-millimeter radius times sin⁡ ⁣(30∘)\sin\!\left(30^{\circ}\right), giving forty millimeters. The two radii and that chord therefore form an equilateral triangle for each of the six sectors. This relationship belongs to a regular hexagon constructed at equal sixty-degree gaps; it is not a rule that every polygon side or circle chord equals its radius.
Why is the circle circumference longer than the polygon perimeter?
The hexagon uses straight chords between its six selected vertices, while the circle follows the curved arcs between the same endpoints. Six forty-millimeter sides total 240mm. The full circle of radius forty has circumference eighty times π\pi, approximately 251.327mm, exceeding that perimeter by approximately 11.327mm. Name the straight and curved paths explicitly rather than using a circle's ring number to report the polygon boundary length.
Does selecting every second spoke produce six or twelve vertices?
The original sheet has twelve distinct spoke directions separated by thirty degrees. Selecting every second one gives six directions separated by sixty degrees: zero through three hundred in the stated sequence. The closing point at a full turn is the original zero-degree point, not another distinct vertex. Unused spokes remain background references. Mark the intended six points and straight connections so a reader does not count all printed intersections as polygon corners.
Are the forty-millimeter radius and eighty-millimeter diameter interchangeable?
The radius measures from the origin to a point on the selected fourth circle. Its diameter spans through the origin between opposite points and is twice as long, eighty millimeters. The regular hexagon's side is forty millimeters for the separately justified sixty-degree chord relationship. Keep these measurements named individually; their relationships do not make a diameter, side, radial interval count, and vertical component the same quantity.
Does the download include the hexagon's sides and formulas?
It generates the chosen blank gray polar references, angular labels, and origin treatment. The selected vertices, straight polygon sides, center connections, and formulas are annotations supplied by the worksheet author. Save the completed comparison in its own master and keep the background address alongside it. A reconstructed lattice alone does not identify the six-point selection or explain why the polygon perimeter differs from the containing circle's circumference.

About This Tool

Letter ten-millimeter gray polar paper with twelve spokes provides a restrained background for a six-point construction on a fixed circle. Selecting alternate directions creates sixty-degree angular gaps while the radial choice remains independent. The author supplies the polygon vertices and straight sides, making their relationship to the circular reference visible rather than assuming that every printed spoke belongs to the hexagon.

The fourth-ring example compares a straight polygon boundary with the curved circle that contains its vertices. A forty-millimeter radius produces a forty-millimeter side for this regular hexagon through the sixty-degree chord relationship. Six sides total 240mm, while the full circumference is longer. The component offsets and diameter describe other measurements of the same construction. Keeping these quantities named prevents a familiar geometric equality from obscuring which path or direction the calculation actually measures.

A retained comparison needs the selected angles, straight-versus-curved outline key, and formulas beside its reproducible gray field. Try a chord endpoint and radius label against the background, preserve full Letter placement when ten-millimeter radial intervals matter, and measure the completed copy. Save that annotated worksheet with its selected settings address. The pair preserves both a repeatable polar reference and the construction explanation needed to distinguish six authored polygon sides from twelve background spokes and a circular circumference.

Original Preset Reference

Polar Paper Letter, 10mm, Gray, 12 Spokes; 10mm circles, Gray Lines polar grid reference for worked hexagon: fourth-ring vertices at sixty-degree increments
Printable polar graph paper for Letter worksheets with 10mm circles, 12 spokes, and gray 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.