# VectorPaint inscribed regular hexagons: verify six sides and six equilateral triangles
Language: en
Canonical: [VectorPaint inscribed regular hexagons: verify six sides and six equilateral triangles]()
中文：[中文 HTML]()
English: [English HTML]()
Published: 2026-11-13T00:00:00+08:00
Modified: 2026-11-13T09:43:51+08:00
Author: Bison
Use a radius-four circumcircle to calculate side, apothem and polygon area without confusing the apothem with side length.
English translation of the [Chinese original](), translated on 2026-11-13. Original publication dates, attribution, code and version notes are retained.
A six-sided outline need not be regular, and six points on a circle still require spacing checks. Draw a radius-four inscribed regular hexagon in VectorPaint, distinguishing centre, vertices and side midpoints. Compute on paper without inventing an automatic proof function.
## Fix the circumcircle and coordinate directions
Use a square canvas with O as a local origin, mathematical x rightward and y upward, at about 80 pixels per unit. Radius 4 means diameter 8, or 640 pixels. The [Illustrative Mathematics inscribed-hexagon task]() supports equal angular spacing and six equilateral triangles; these numerical examples are calculated independently. If screen y increases downward, reverse its mapping rather than placing mathematical upward points at larger screen y values.
## List six vertices in boundary order
Starting at the rightmost point and moving counterclockwise, use A=(4,0), B=(2,2√3), C=(−2,2√3), D=(−4,0), E=(−2,−2√3), F=(2,−2√3). With √3 approximately 1.732, the upper/lower rows are approximately ±3.464. Retain exact radicals in notes and mark approximate entry or manual placement. Replacing the six points with an evenly spaced rectangular grid for easier selection destroys the circumcircle and side relationships. Check points before connecting them.
## Close the boundary without making a star
Connect A–B–C–D–E–F–A along adjacent points rather than skipping points into a star. Each side is 4: AB has differences −2 and 2√3, squared sum 4+12=16 and length 4. Centre-to-vertex distances are also 4, a property of this regular hexagon rather than every regular polygon. Mark vertex centres in an audit copy if thick strokes obscure them; do not connect edges to marker rims and create small gaps.
## Connect vertices to O to obtain six triangles
On a separate layer connect O to all vertices. Adjacent rays span 60 degrees, with six making 360. Each sector triangle has two radius-four sides and a side of length four, so it is equilateral rather than merely similar-looking. A ray accidentally ending at a side midpoint creates different subdivisions. Coloured triangles must share O without dragged gaps if they are said to fill the polygon. Explain whether subdivision lines remain in the finished diagram.
## Distinguish apothem from circumradius
BC is the upper horizontal side, with midpoint (0,2√3). Its perpendicular distance from O is 2√3, approximately 3.464: the apothem, shorter than circumradius 4. Do not relabel it as side length. Use a slant radius to B marked 4 and a vertical perpendicular to BC’s midpoint, distinguished by words as well as colour. Move labels out of a crowded diagram rather than shifting vertices. An arbitrary slanted screen segment is not the required perpendicular distance.
## Compute one triangle before comparing with the circle
One triangle has area 4×2√3/2=4√3, so all six have 24√3, approximately 41.569 square units. The circumcircle has area 16 times pi, approximately 50.265, making the polygon about 82.7% of the disk. A length ratio cannot replace this area ratio, and a circumference is not the sum of six chords. Separate exact expressions and rounded values, using consistent rounding. Pixel discrepancies cannot justify changing valid paper calculations. Independently, perimeter is six times four, or 24. Multiplying by apothem 2√3 and dividing by two gives the same 24√3 area. The 24 in perimeter has length units; the area coefficient has a different role. Equal-looking numbers do not make perimeter equal area. Retain this second calculation and locate every quantity in the diagram.
## Troubleshoot unequal spacing and elliptical scaling
After export check the A–D diameter, parallel upper/lower sides, all six sides and radii. Horizontal stretching turns the circle into an ellipse and breaks equal-side/equilateral relations; changing only the shape name does not preserve the numbers. Repair one displaced vertex and its two edges instead of moving the other five to accommodate it. Refine strokes and labels after geometry, retaining audit circles and axes. Unopened export formats remain unverified.
## Reopen and verify vertices, apothem and area
Save, leave and reopen, tracing A through F back to A. Point separately from O to a vertex and side midpoint, explaining 4 versus 2√3. Retain a radius, apothem and area units in delivery, with guides hidden or explained rather than mistaken for extra perimeter edges. Open final output and check radicals and approximations for readability. Another reader should reconstruct the six-triangle calculation and recognise placement limitations; this is no automatic validation of every angle or manufacturing precision.
## Related tutorials and sources
[VectorPaint triangle medians: check midpoints and the two-thirds centroid position](), [VectorPaint hex-nut diagrams: separate six-sided outline, circular hole and section convention]().
Checked on November 13, 2026: VectorPaint public version 1.0.8, requiring iOS / iPadOS 17.0 or later. Scope follows the [official App Store]() and [Apple public lookup](). Manual exercises use controls actually available in your installed version; external calculations, written records and diagrams imply no built-in automatic analysis. No step-by-step physical-device, private-file or all-export-environment testing is claimed.