Reasoning · Grade 3-1 Shapes and Length

Problem

Perimeter of a figure built from squares

Seven 8 cm squares stand as diamonds in a row, each overlapping its neighbour by half. Only the edges left on the outside are red; the hidden parts are not. Top and bottom mirror each other. Find the total length of the red line.
8 cm
Your answer
How to solve
Strategy Draw a Diagram — The red line is a path along the edges of the diamonds, so drawing and tracing it edge by edge makes clear which pieces show. Splitting the outline into the top zigzag and the bottom zigzag (subproblems) and noticing the repeating peak-valley pattern lets us add up the visible edge pieces without measuring anything diagonal.
1STEP 1

Trace the leftmost diamond

The leftmost two edges overlap nothing: a full 8 cm each.

8 + 8 = 16 cm at the left end
2STEP 2

See the repeating peak-to-peak unit on top

With half overlaps, one peak-to-peak unit is 8 cm.

4 + 4 = 8 cm per peak-to-peak unit
3STEP 3

Add up the whole top zigzag

The top runs 8 + 6 × 8 + 8 = 64 cm.

8 + 6 × 8 + 8 = 8 + 48 + 8 = 64 cm
4STEP 4

Use the mirror symmetry for the bottom

The bottom matches, so in all 128 cm.

64 + 64 = 128 cm
Answer
128 cm
64 × 2 = 128
Each diamond contributes 4 edges of 8 cm = 32 cm, and 7 separate diamonds would total 224 cm. Each of the 6 overlaps buries exactly 2 full edge-lengths (16 cm), hiding 6 x 16 = 96 cm. Then 224 - 96 = 128 cm, matching the answer. The units are cm of length, which is correct for an outline.
Takeaway

Going down half an edge and up half an edge adds up to one whole edge, so the zigzag is just neat counting of 8 cm pieces!

  • Trace the leftmost diamond
  • See the repeating peak-to-peak unit on top
  • Add up the whole top zigzag
  • Use the mirror symmetry for the bottom