Free printable resources · Original practice
Choose flat-top or pointy-top hexagons at the same 10 mm across-flats size. Trace a seven-cell flower and calculate its perimeter and area.
Choose flat-top or pointy-top hexagons at the same 10 mm across-flats size. Trace a seven-cell flower and calculate its perimeter and area.
Choose US Letter or A4, then download the PDF for printing or the SVG for editing. Keep the student worksheet separate from its answer or reference completion. All files, including the answers, are free without signup.
Original blank printable. Choose Letter or A4; print at actual size.
Original blank printable. Choose Letter or A4; print at actual size.
Take one cell and all six edge-sharing neighbors. There are 7 × 6 = 42 cell-edge incidences. Each of the 12 shared internal edges was counted twice. Remove those 24 incidences to leave 18 edges on the outside boundary. The center touches six neighbors, and the surrounding ring contributes six more shared pairs.
| Quantity for a 10 mm across-flats grid | Result |
|---|---|
| Single side s | 10/√3 mm ≈ 5.774 mm |
| Flower perimeter · 18s | 60√3 mm ≈ 103.923 mm |
| One regular hexagon area | 50√3 mm² |
| Seven-cell area | 350√3 mm² ≈ 606.218 mm² |
The worksheet enlarges the cells to 28 mm across flats for clear edge counting. Its numeric questions still specify the 10 mm paper size. Treat the large illustration as a schematic and use the stated scale for the arithmetic.
Original practice with a separate reference answer. Choose Letter or A4.
Flat-top paper has a horizontal top edge. Pointy-top paper has an upper vertex. Turning the grid changes its orientation, not the side length or across-flats definition. Only whole hexagons are drawn within the reserved region. Neighboring cells share a single vector edge, keeping internal lines the same weight.
The grid size is defined across opposite parallel flats: 10 mm. It is not a 10 mm side length. First print the matching Letter or A4 PDF at Actual size / 100%, then measure the calibration bars and a flat-to-flat span on the physical sheet. A regular hexagon at this size has side length 10/√3 mm, about 5.77 mm, and opposite-corner distance 20/√3 mm, about 11.55 mm. Use those separate definitions when comparing a ruler measurement or editing the SVG. A monitor preview alone cannot establish the printed size.
Determine whether your printed sheet features flat-top or pointy-top hexagons by examining the edges. Flat-top hexagons display horizontal edges at the top and bottom, while pointy-top hexagons present vertices at these positions. This orientation affects how you visualize the seven-cell flower pattern. Ensure you are working within the inset area where full hexagons are visible, staying away from the page margins; these files deliberately include only whole cells. Correct orientation helps in accurately tracing the shared internal edges and outer boundaries. Select the orientation that best suits your tracing needs, as both provide the same geometric properties for the central cell and its six edge-sharing neighbors.
Locate a central hexagon and trace its six immediate edge-sharing neighbors to form the seven-cell flower shape. This configuration creates twelve shared internal edges and eighteen outer edges. The diagram provided is enlarged for readability and marked as schematic, so focus on the topology and use the stated 10 mm size for the arithmetic rather than measuring the enlarged illustration. Count the eighteen outer edges to verify your tracing. At the ten millimeter flat-to-flat scale, the total perimeter of this shape is approximately one hundred three point nine two millimeters. The total area of the seven cells is roughly six hundred six point two two square millimeters. Use these values to check your manual measurements against the theoretical geometric constants.
| Watch for | Check instead |
|---|---|
| Measuring across corners instead of flats | Always measure the distance between opposite parallel flat edges, which must be ten millimeters. Measuring corner-to-corner yields eleven point five five millimeters, leading to incorrect scaling. |
| Tracing only the center and five neighboring cells. | A complete flower contains the center plus all six edge-sharing neighbors. Check seven cells, 12 shared edges and 18 exterior edges. |
| Assuming colors are required for answers | Coloring is optional. The correct answer relies on identifying the boundary and counting edges. Focus on the geometric structure of the seven cells rather than applying specific colors to the diagram. |
Use the PDF that matches the paper loaded in your printer. Choose Actual size / 100% and turn off Fit, Shrink and browser headers. The SVG keeps millimeter dimensions, but an editor or browser can rescale it; the PDF is the simpler print path. Measure any calibration bars with a separate physical ruler when the exercise depends on printed size. A screen preview does not establish physical scale.
Packs combine the same individual pages; they are not extra different exercises. Print a student copy first and keep references nearby for review. Letter and A4 preserve the stated grid spacing or model definition while their surrounding white space can differ. Color is optional: the references also use line position, text or relative lightness to explain the result.
The activity enlarges each hexagon to 28 mm across flats to make edge counting clear. It keeps the same regular geometry. Calculate the requested result using the specified 10 mm grid, not the enlarged diagram.
The exact perimeter is 18 × (10/√3) = 60√3 mm. This is approximately 103.923 mm; a finite decimal is a rounded result. Seven hexagons have exact area 350√3 mm², approximately 606.218 mm².
The seven-cell flower configuration has twelve shared internal edges. These are the lines where the central hexagon touches its six neighbors, and where neighbors touch each other within the cluster.
Yes. Every linked PDF, SVG, worksheet and answer is free without signup. The original Zumola resources use CC BY 4.0: credit Zumola, link the license and indicate changes. Analytics consent is not required to download or use them.
A regular hexagon has six equal sides and six vertices. Across-flats means the perpendicular distance between two opposite flat sides; across-corners means the distance between opposite vertices. For a regular hexagon, across-corners is larger than across-flats, so a sheet described only as 10 mm is ambiguous unless the measurement is named.
In a pointy-top layout, one vertex points up and the flat sides face left and right. Neighbours sit above, below, and diagonally. In a flat-top layout, one flat side faces up and vertices point left and right. Neighbours sit left, right, and diagonally. The hexagons themselves are congruent; only the arrangement rotates.
When you switch orientation, keep the across-flats size fixed if you want comparable cell areas. Rotating the layout does not change a single hexagon's area, but it does change which rows and columns appear straight on the page.
Hex grids are often labelled with three coordinates, sometimes called cube coordinates, where the three values sum to zero. If you label a cell (1, -1, 0), its neighbour across one edge might be (2, -2, 0). The sum-to-zero rule is a consistency check: if a label's three parts do not add to zero, the label is wrong for that coordinate system.
A simpler alternative is axial labelling with two numbers plus a fixed rule for the third. Either way, decide the rule before labelling and write it at the top of the sheet. Mixing two labelling schemes on one sheet is the most common cause of mismatched neighbours.
Labels are for reference, not for drawing. Place them inside the hexagon near one corner, or outside along a margin, so they do not obscure the edges you are tracing.
A correct tiling has no gaps and no overlaps: every interior edge belongs to exactly two hexagons, and every vertex where three hexagons meet is shared by three. You can test this without measuring by tracing one edge and counting how many hexagons touch it.
If an edge is touched by only one hexagon, you have a gap or a boundary. If an edge appears to belong to three hexagons, you have an overlap or a drawing error. Both are visible by eye on a printed sheet.
This check also catches orientation mistakes. A pointy-top hexagon placed in a flat-top row will fail the shared-edge count because its sides will not align with its neighbours.
A sheet uses 10 mm across-flats pointy-top hexagons. Label the centre cell (0, 0, 0) and its six neighbours using three-part coordinates that sum to zero. Then verify one shared edge.
The centre is (0, 0, 0). Six valid neighbours are (1, -1, 0), (1, 0, -1), (0, 1, -1), (-1, 1, 0), (-1, 0, 1), and (0, -1, 1). Each triple sums to zero: for example, 1 + (-1) + 0 = 0. Pick the edge between the centre and (1, -1, 0). That edge is shared by exactly those two hexagons, so the count is two. Pick the vertex where the centre, (1, -1, 0), and (1, 0, -1) meet; three hexagons share it, which is correct. If either count differs, the label or the placement is wrong.
No. A regular hexagon's area depends on its side length, not its orientation on the page. Rotating the layout changes neighbour directions and page alignment, but each cell keeps the same area if the across-flats size is unchanged.
The sum-to-zero rule is a bookkeeping convention that keeps labels consistent across the grid. If the three parts do not add to zero, the label does not correspond to a valid cell in that coordinate system, so neighbour relationships become unreliable.
Three. In a regular hexagonal tiling, interior vertices are shared by exactly three hexagons, and interior edges by exactly two. Counting these is a quick way to spot gaps, overlaps, or a misplaced cell.
