Justify why the plane cannot be tiled with a regular pentagon. (Hint: Read the first 3 sentences of ‘Think and Reflect’ in the section on tiling.) (There are many ways to tile the plane using a suitable irregular pentagon. The most recent method was found in 2015.)
Step-by-step solution
Idea: In a tiling, the angles round every vertex add up to 360°. Check whether copies of 108° can ever make 360° (or 180° next to a straight edge).
- Interior angle of a regular pentagon = (5 − 2) × 180°5 = 540°5 = 108°.1 mark
- At a vertex where only corners of pentagons meet, the angles must add up to 360°: k × 108° = 360° gives k = 3⅓, not a whole number. Three pentagons leave a gap of 36°; four overlap by 72°.1 mark
- If a corner touched the middle of another pentagon’s edge, the corners there would have to make 180°: k × 108° = 180° gives k = 1⅔, again impossible. So no arrangement works: the plane cannot be tiled with regular pentagons.1 mark
Check: 3 × 108° = 324° (gap of 36°); 4 × 108° = 432° (overlap of 72°) ✓. For comparison, squares (90° × 4) and regular hexagons (120° × 3) do tile.
Answer to write in the exam
Each angle of a regular pentagon = (5 − 2) × 180° ÷ 5 = 108°
Angles round a vertex of a tiling = 360°; k × 108° = 360° ⇒ k = 3⅓ (not a whole number)
Corner on another tile’s edge: k × 108° = 180° ⇒ k = 1⅔ (not whole)
∴ Regular pentagons cannot tile the plane.
Common mistakes that cost marks
- Using 72° (the exterior angle) as the corner angle. The angle at the corner of the tile is 108°.
- Forgetting the case where a corner meets the middle of an edge; checking it makes the proof complete.
- Thinking no pentagon can tile. Many irregular pentagons can; only the regular one fails this angle test.
How this can come in the exam
Which regular polygon can tile the plane on its own?
- Regular pentagon
- Regular hexagon
- Regular octagon
- Regular decagon
Show answer
(B) Regular hexagon
Its angle is 120°, and 3 × 120° = 360°. The others have angles 108°, 135°, 144°, none of which divides 360° exactly.
Try one yourself
Can regular octagons tile the plane on their own? What extra shape is used on floors with octagonal tiles?
Show answer
No: each angle is 135° and 360 ÷ 135 is not whole (two octagons leave 90°). Small squares fill the 90° gaps.
More questions like this
- Draw a non-convex 4-gon DART. Show how we can tile the plane with copies of DART. Both methods that we discussed earlier will work. Which do you prefer?
- Using a fact about parallelograms, show how to tile the plane using any given triangle. (Hint: Can you use the parallelogram tiling in the introduction?)
- Mark the midpoint of the line drawn on the paper (see the figure), given that the horizontal lines are equally spaced. Justify your answer.
- You know that the sum of angles of a quadrilateral is 360°, even for a non-convex quadrilateral. (Recall the proof.) Now consider a self-intersecting quadrilateral ABCD, where AB and CD intersect at point E. Show that ∠A + ∠B + ∠C + ∠D < 360°. Can you construct ABCD such that ∠A + ∠B + ∠C + ∠D = 2°?
- In a parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see the figure). Show that APCQ is a parallelogram.
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