Ruiya Math Blog · No. 6 · One Thinking Move
For perimeter, trace the fence—not the wallpaper lines.
A diagram can contain many visible lines. Perimeter uses only the outside edge. The important question is not ‘can I see this line?’ but ‘does this line touch the outside?’
A Ruiya-original PSLE-style practice guide for Primary 5, Primary 6 and PSLE
The first useful observation
A shared edge is inside the joined shape. It may be drawn, but it is not part of the outside boundary.
Three 6 cm triangles have 9 sides altogether. Two shared sides are inside the shape, so 4 counted sides disappear: 9 − 4 = 5 exposed sides.
The challenge
The joined-triangle garden challenge
- • Three identical equilateral triangular garden beds are joined edge-to-edge in a zig-zag chain.
- • Each side is 6 cm long.
- • The first triangle shares one full side with the second. The second shares another full side with the third.
- • All three beds together form one shaded shape. Find its perimeter.
This is a Ruiya-original analogue, not a past-paper question.
Why this strategy fits
Use a picture your brain already understands.
If three neighbours join their gardens, the fences between neighbours do not surround the combined garden. Only the fence that faces the street remains in the outside perimeter.
Count every side first. Three triangles have 3 × 3 = 9 sides before joining.
Remove the internal edges twice. Each shared side was counted once for each triangle, so each shared edge removes 2 counted sides. Two shared edges remove 4 sides.
Measure the exposed boundary. There are 9 − 4 = 5 outside sides. Each is 6 cm, so perimeter = 5 × 6 = 30 cm.
Check your answer
Walk your finger only around the outside of the joined figure. You cross five 6 cm edges and never cross either shared edge.
Answer: The perimeter is 30 cm.
Traps to watch for
The usual wrong turn is often a very reasonable first thought.
Counting every visible line
Internal dividing lines are wallpaper lines: visible, but not on the fence.
Removing a shared side only once
It was counted in both triangles, so it must be removed twice from the total side count.
Mixing area and perimeter
Perimeter is a length around the edge. It is measured in cm, not cm².
How an examiner can change it
Keep the thinking move, not the story.
The picture may use shaded triangles, semicircles, tiles or composite shapes. Mark each boundary edge with a small tick as you trace it. If an edge has shape on both sides, it is inside and does not count.
Try a fresh variation
Four identical equilateral triangles of side 5 cm form a chain, with three shared full sides. What is the perimeter of the combined shape?
Do not look for the same numbers. Ask which relationship is still doing the work.
Outside the exam
A useful model helps you make a fair decision.
Builders estimate fencing, edging and safety rails from exposed boundaries. They do not buy fence material for a wall shared inside a combined courtyard.
Try it with Ruiya
Want help choosing the first step?
Send Ruiya a hard question and ask: “What is the relationship I should notice before I calculate?”
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