Equal area, different boundary

Count Exposed Edges When Area Stays Fixed

Record area and boundary in different columns

In Equal area, different boundary, twelve identical squares provide twelve square units of area, however you rearrange them without overlap. The boundary measures exposed unit-length edges instead. Give the record separate headings for square units and length units. A longer-looking arrangement is not automatically larger in both columns. Before comparing answers, count the occupied cells and check that the pieces form one edge-connected shape. A diagonal touch alone does not meet this activity's connection rule. Keep the examples free of enclosed holes so the boundary being traced is easy to follow.

Begin with a long strip as a fresh comparison

Put all twelve squares in a straight row. Its two long sides each contribute twelve length units, and its ends contribute one each, giving a boundary of twenty-six. It still covers twelve square units. Compare that with the printed three-by-four rectangle, which covers the same area but has fourteen boundary units. The difference is not missing paper. Many edges that were exposed in the strip become shared inside the compact rectangle. Ask participants to point to a shared edge before attempting a formula, so the change has a visible explanation.

Count each shared edge in the right role

If you count four sides for every tile, twelve tiles produce forty-eight side appearances. An internal shared edge appears twice in that count, once for each neighbouring tile, but neither appearance belongs to the exposed boundary. Subtract two for each shared edge. The strip has eleven shared edges, giving forty-eight minus twenty-two, or twenty-six. The three-by-four rectangle has seventeen shared edges, giving forty-eight minus thirty-four, or fourteen. This supplies a second way to check the tracing, rather than a rule to apply without first deciding which tiles actually share a full side.

Move one tile while keeping all twelve

Label the rectangle's columns one to four and rows one to three. Remove its upper-right tile at column four, row three. Then attach that tile at column five, row one, beside the bottom-right tile. The resulting shape remains connected and has twelve occupied cells. Its boundary is sixteen. Removing the corner lost two shared edges; reattaching the tile created one new shared edge. There are now sixteen shared edges in total. Trace the new outline as well as checking the arithmetic, since a misplaced tile could create a different arrangement from the one described.

Compare a shape that changes without changing boundary

Build an L using seven tiles in a horizontal row and five more rising from the leftmost tile. There are twelve tiles altogether: the corner is counted only once. This shape also has eleven shared edges, so its boundary is twenty-six, the same as the straight strip. It is a useful warning against deciding by appearance alone. A change of shape need not change the perimeter, just as a fixed area need not fix it. Describe the actual construction beside the numbers so another group can rebuild and verify the comparison.

Separate a measured pattern from a global claim

The NRICH area-and-perimeter investigation explores how these two quantities can vary independently. For this twelve-tile activity, the checked examples show several possibilities, not a proof that every other arrangement has a greater or smaller boundary. If someone says the rectangle is best, ask what “best” means and which arrangements have been considered. You may record a conjecture for later investigation while still completing the current task. A clear observation with limits is more useful than an unsupported minimum announced because the shape looks compact.

Finish with a trace someone else can follow

Have the builder draw the occupied cells, the tracer mark exposed edges, and the checker recount independently. Rotate those roles for the next arrangement. Use a dot or small tick on each edge already counted so none is missed or counted twice. If the totals disagree, compare a particular section of boundary instead of recounting different routes aloud simultaneously. Keep the tile size fixed throughout the comparison and use squared paper if cutting is inconvenient. End with two same-area shapes and justified boundary counts, retaining any extra examples as evidence rather than as a new scoring contest.