Can we use any given quadrilateral to tile the plane? If not, which quadrilaterals can be used and which cannot?
Step-by-step solution
Idea: Tiling means covering the plane with copies of a shape with no gaps and no overlaps. Two facts make every quadrilateral work: (1) a half-turn about the midpoint of a side swaps the two ends of that side, so the copy shares the whole side; (2) the four angles of a quadrilateral add up to 360°, exactly the full turn needed round each corner.
- What must happen at a corner. Where corners of tiles meet, the angles around that point must add up to exactly 360°, otherwise there is a gap (less than 360°) or an overlap (more than 360°).½ mark
- The angle sum fits. The angles of any quadrilateral add up to 360° (a diagonal splits it into two triangles, 180° + 180°). So if one copy of each angle meets at every corner, the corner is filled exactly.1 mark
- How to place the copies. Take a copy and turn it through 180° about the midpoint of one side. The two ends of that side swap places, so the new copy lies along the whole side with no gap. Do this for every side of every copy you add.1 mark
- Conclusion. This works for every quadrilateral, even an irregular one or one with a dent. Rectangles and parallelograms are just the easy cases where all copies point the same way.½ mark
Check: Try it with paper: cut 8 copies of an odd-shaped quadrilateral, number the angles 1, 2, 3, 4 on each, and place them so that each corner gets one angle of each number. They lock together with no gaps.
Answer to write in the exam
Sum of angles of any quadrilateral = 360°
Turn a copy through 180° about the midpoint of a side ⇒ the copy fits exactly along that side
Repeat for every side ⇒ at each corner the four angles meet once: total 360° ⇒ no gap, no overlap
∴ Every quadrilateral (convex or not) can tile the plane; none is excluded.
Common mistakes that cost marks
- Answering “only squares, rectangles and parallelograms”. Every quadrilateral tiles; the copies just need to be turned, not only slid.
- Thinking a quadrilateral with a dent (a reflex angle) cannot tile. It can: its angles still add up to 360°.
- Mixing up quadrilaterals with other polygons: a regular pentagon cannot tile, because 108° does not divide 360°.
How this can come in the exam
Four copies of a quadrilateral meet at one point of a tiling, one copy of each angle. The angles around that point add up to
- 180°
- 270°
- 360°
- 720°
Show answer
(C) 360°
The four angles of a quadrilateral add up to 360°, which is exactly one full turn.
A floor is to be tiled with identical tiles in the shape of a quadrilateral with angles 70°, 95°, 110° and 85°. Explain why this is possible at the corners of the tiles.
Show answer
70° + 95° + 110° + 85° = 360° (1 mark). If one angle of each size meets at every corner, the angles fill the full turn of 360° with no gap or overlap; placing copies by half-turns about midpoints of sides achieves this (1 mark).Try one yourself
Can the plane be tiled with copies of a kite? Give a reason.
Show answer
Yes. A kite is a quadrilateral, its angles add up to 360°, and half-turned copies about the midpoints of its sides fit together with one copy of each angle at every corner.
More questions like this
- Informally, a quadrilateral is a figure with four straight sides, as in the first figure ABCD in the figure below. But consider the other six figures in the figure: the five plane figures NOPE, SILY, DART, CUTS, OPENS and the non-planar BENT. Should we call all these figures quadrilaterals? If you answer ‘no’ for any of them, how will you define a quadrilateral so that such a figure is excluded? As you can see, some care is needed to precisely define what we think of as a quadrilateral.
- To prepare, let us first consider how we can define a triangle. Let A, B and C be three points. Can we say that ∆ABC consists of points on the three line segments AB, BC, CA?
- Can we similarly define a quadrilateral ABCD?
- Take any 4 distinct non-collinear points A, B, C and D. Will segments AB, BC, CD and DA always form a quadrilateral as we visualise it?
- Can you come up with a definition that rules out self-intersecting quadrilaterals like CUTS?
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