Any four-gon covers a floor
Show on a big screenFour copies of any quadrilateral can meet without a gap. Each copy is a tile, a flat shape for covering a floor.
- One quadrilateral tile, corners A, B, C, D.
- Turn a copy half a turn about the midpoint of side AB.
- The turned copy shares the whole side AB with the first.
- The two tiles meet with no gap along the seam.
The rule
Corners that meet at a point must total 360°, a full turn. The four corners of any quadrilateral already total 360°. Cut the quadrilateral along a diagonal inside it. That gives two triangles. Their angles total 180° + 180° = 360°.
Rule: Four copies of one quadrilateral meet at a point. Each copy gives one corner. There is no gap and no overlap.
Some shapes need a different number of copies. Regular means all sides and angles are equal. A regular hexagon has 120° corners. Three copies meet at a point. But a regular pentagon has 108° corners. Three copies leave a gap of 36°.
Any four-sided shape works. It does not need equal sides.
| Tile | Angles | Total |
|---|---|---|
| A tidy rectangle | 90°, 90°, 90°, 90° | 360° |
| A pushed-in tile | 40°, 75°, 210°, 35° | 360° |
The half-turn helps build the floor. The tile and its half-turned copy join into one bigger shape. Its opposite sides are equal and parallel. Copies of that bigger shape fit side by side.
Add the four corners round a point.
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Try the lab to finish this lesson.