VectorPaint annulus diagrams: distinguish both radii from equal ring width

English translation of the Chinese original, translated on 2026-11-13. Original publication dates, attribution, code and version notes are retained.

An annulus is not a circle whose radius equals the ring width. Make two same-scale region diagrams in VectorPaint, subtracting the inner disk from the outer one before comparing equal widths with different areas. Check mathematics separately from placement; filling a shape does not verify a calculation.

Fix units, scale and a shared centre

Use an 1800×1200 canvas and declare 80 pixels per abstract unit, leaving centres and radius-label space in both panels. Ring A has outer radius 5 and inner radius 3; B has outer 4 and inner 2, not centimetres. The OpenStax area chapter supplies the circle-area relationship; this original subtraction example is computed independently. Keep one scale and centre references. Enlarging B for layout would invalidate comparison of displayed areas.

Measure both radii from the same centre

A’s outer and inner diameters are 10 and 6 units, or 800 and 480 pixels. Align both circle centres to a cross before labels; left-edge alignment is not concentricity. If precise dimensions are unavailable, label the drawing approximate while calculating with 5 and 3. An offset inner circle creates varying width and no longer depicts this concentric example. Repair the centre rather than using shading to explain the offset as material texture.

Decide hole semantics before a filling method

If your installed version offers compound paths or suitable shape operations, test a copy to verify an actual hole; do not invent menus. A background-coloured small disk can make a white-background teaching diagram, but label it as an overlay rather than promising transparency. Place a coloured or checked backdrop behind it: a persistent white disk is an object, not a hole. The mathematical area does not become an area of white paint; explain representation separately from region definition.

Use a difference of squared radii

A’s outer disk has area 25 times pi and its inner disk 9 times pi, giving 16 times pi, approximately 50.265 square units. Width 5−3 is 2, but squaring 2 gives a different radius-two disk with area 4 times pi. Label correct and wrong calculations side by side: one removes the entire inner disk, while the other computes another disk. Calculate before labels; counting PNG pixels cannot yield an exact pi-based area, and antialiasing does not define a mathematical boundary.

Calculate the second width-two ring independently

B also has width 4−2=2, but disk areas 16 and 4 times pi yield 12 times pi, approximately 37.699. A exceeds B by 4 times pi. Equal radial width does not imply equal area. At the same scale B’s diameters are 640 and 320 pixels. If different scales are used for layout, withdraw direct visual-area comparison and label the scales, retaining calculated values. Layout convenience cannot establish equality. Along a rightward radius, the inner boundary is at 3 and outer at 5, giving width 2. The distance 10 across the whole outer disk is its diameter, not another ring width.

Keep boundary distances and area ratios separate

For A, outer and inner boundary distances from O are 5 and 3, and intermediate points lie between them. Draw both boundaries without pursuing boundary-inclusion effects on area. The ring occupies 16/25=64% of the outer disk, while the inner disk occupies 36%, totalling 100%. Width/radius=2/5=40% is a length ratio, not the area ratio. Attach leaders to the appropriate radii and keep percentage labels with region descriptions, not where 64 might be read as diameter or pixels.

Troubleshoot white overlays and off-centre holes

Open the saved PNG on white and dark backdrops. If a white inner disk hides the dark background, retain its overlay status or use a genuinely available hole operation and verify again; PNG alone guarantees no transparency. Uneven width calls for centre checks; wrong values call for radius/diameter checks. Treating diameter 10 as radius 10 changes the area scale and cannot be repaired by relabelling alone. Hide ordinary guide objects for delivery while retaining an annotated audit copy and the original.

Reopen and verify both rings and calculations

Close and reopen the canvas, checking A’s 5/3 and B’s 4/2 radii and shared units before reading 16 and 12 times pi. Preserve overlay-versus-hole labelling, correct radius leaders and the scope of A’s 64% outer-disk fraction. Open final output and inspect centres and labels reduced, leaving untested formats unverified. Another reader should reconstruct the subtraction and explain unequal areas with equal widths; this is no automatic annulus solver or material-quantity calculation.