Perimeter of Complex Shapes
Warm-up
Project an L-shaped swimming pool deck with only some sides labelled and announce: "The railing company charges by the metre. Quote the job." Let pairs hit the wall: two side lengths are missing.
Surface the rescue idea from the room: opposite sides of a rectilinear shape must account for each other — the missing pieces are findable from the given ones. Today is about perimeter as reasoning, not adding.
Explore
Missing-sides workshop: three rectilinear figures (L, T, and staircase shapes) with strategic gaps in the labels. Pairs find every missing side by matching horizontal runs against horizontal runs and vertical against vertical, then compute each perimeter.
Then the surprise investigation: cut a rectangular notch out of a rectangle's corner and re-walk the boundary. The perimeter DOESN'T change (corner notch: the two new edges replace equal removed lengths)… but a notch cut into the middle of a side ADDS perimeter. Pairs test both, measure, and explain the difference with arrows along the walk.
Formalize
Formalize the accounting principle for rectilinear shapes: total left-facing edges equal total right-facing edges; total up equals total down. Perimeter is the full boundary walk:
The corner-notch invariance, stated once cleanly: pushing a corner inward relocates boundary without changing its total length — the two new segments equal the two they replaced. Mid-side notches add new boundary with no compensation. Location of the cut, not the cut itself, decides.
Practice
Practice: three missing-side figures of increasing nastiness; one perimeter-preserving redesign challenge ("change this shape's area without changing its perimeter"); one metric problem with mixed units (cm segments summed into metres).
Exit ticket: an L-shape with four of six sides labelled — find the perimeter and mark which sides you deduced.
Exit ticket
Practice: three missing-side figures of increasing nastiness; one perimeter-preserving redesign challenge ("change this shape's area without changing its perimeter"); one metric problem with mixed units (cm segments summed into metres).
Exit ticket: an L-shape with four of six sides labelled — find the perimeter and mark which sides you deduced.
The deck: going clockwise from the top-left — top 12 m, right side 3 m, inner step across 5 m, inner step down 4 m, then the unlabelled bottom and left sides.
Step 1: Deduce the bottom: horizontal runs right must equal horizontal runs left. Top runs 12; the step-across runs 5 the other way… track signs by direction: rightward total = leftward total → bottom m.
Step 2: Deduce the left side: downward = upward → left side m.
Step 3: Walk and sum: m of railing.
Step 4: Audit: an L has 6 sides; six numbers were summed ✓, and each deduced side is positive and sensible against the picture.
The commission: a 6 m × 6 m garden (P = 24 m, A = 36 m²) must keep its 36 m² but present AT LEAST 30 m of edge for flower borders.
Step 1: Confirm the square's failure: 24 m of edge — 6 short.
Step 2: Stretch: try 9 × 4 (A = 36 ✓): . Closer. Try 12 × 3: ✓ exactly. Try 18 × 2: — overshoot available if wanted.
Step 3: Or go rectilinear: a 6×6 with a 2×2 notch pushed into one side mid-edge: area drops to 32 ✗ — the notch spends area. Compensate: start from 38 m²… simplest honest answer stays rectangular: 12 × 3.
Step 4: Present the trade-off table (shape, area, perimeter) and the recommendation with reasons. Design constraints turn perimeter from an exercise into a negotiation — which is what it is in the world.
The figure: a 3-step staircase shape, total width 9 m, total height 6 m, each step 3 m wide and 2 m tall.
Step 1: The insight that skips all deduction: slide every horizontal step-edge UP to the top — together they span exactly the width, 9. Slide every vertical step-edge LEFT — together they span the height, 6.
Step 2: So the staircase's perimeter equals the BOUNDING RECTANGLE's: m — steps or no steps.
Step 3: Verify by honest walking: three times down the stairs () + bottom + left side = ✓.
Step 4: State the scope of the trick: it works for staircases that only step DOWN-and-RIGHT (monotone). A staircase with a backtrack (an overhang) adds unmatched segments — test one and watch the equality break. Every good shortcut comes with a warranty card listing what voids it.