Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea:Tiling covers a whole surface with copies of shapes that fit together perfectly.
Common stuck point:The procedure for tiling intuition is the easy part; the trap is allowing tiny gaps. Asking "Do the shape's corners meeting at a point add to exactly 360° with no gap or overlap?" first is what keeps a correct-looking calculation from being attached to the wrong concept.
Sense of Study hint:Ask: Do the shape's corners meeting at a point add to exactly 360° with no gap or overlap?
Worked Examples
Example 1
easy
Can regular pentagons tile the plane by themselves (no gaps or overlaps)? Explain using interior angles.Interior angle of a regular pentagon = 108°; does 360° ÷ 108° give a whole number?
Answer
No — regular pentagons cannot tile the plane because 108° does not divide 360° evenly.
First step
1
Step 1: Interior angle of a regular pentagon: 5(5−2)×180°=5540°=108°.
Full solution
2
Step 2: For a tiling to work without gaps, the angles meeting at each vertex must sum to exactly 360°.
3
Step 3: 360°÷108°=3.33..., which is not a whole number. So pentagons cannot fit evenly around a vertex.
4
Step 4: Therefore, regular pentagons cannot tile the plane.
For regular polygons to tile the plane, their interior angle must divide 360° exactly. Only equilateral triangles (60°), squares (90°), and regular hexagons (120°) satisfy this among regular polygons, as 360/60=6, 360/90=4, 360/120=3.
Example 2
medium
A kitchen floor 300cm×240cm is tiled with 20cm×20cm square tiles. How many tiles are needed? If tiles cost $2.50 each, what is the total cost?Kitchen floor: 300 cm × 240 cm. How many 20 cm × 20 cm tiles are needed?
Example 3
medium
A kitchen floor is 4m×3m. Tiles are 25cm×25cm. How many tiles are needed?
Example 4
medium
Can you mix octagons (corner 135°) and squares (corner 90°) to tile a floor? Use the corner test.
Example 5
medium
A patio uses 50cm×50cm tiles. The patio is 6m long and 4m wide. How many tiles?
Example 6
hard
A floor 5m×3m uses two tile sizes: 50cm squares for the border (one row around), and 25cm squares for the inside. How many border tiles?
Example 7
hard
A regular tile fits at a corner with 4 copies. What is its corner angle?
Example 8
challenge
A floor is tiled with both hexagons and triangles. Each corner has 2 hexagons and 2 triangles. Does it satisfy the corner test?
Practice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easy
Which of these regular polygons can tile the plane on their own: equilateral triangle, regular hexagon, regular octagon? Explain using interior angles.Regular hexagon: interior angle = 120°; three meet at each vertex (360° ÷ 120° = 3).
Example 2
medium
A wall 180cm×120cm is covered with rectangular tiles 15cm×10cm. How many tiles are needed? Verify using areas.Wall: 180 cm × 120 cm. How many 15 cm × 10 cm tiles are needed?
Example 3
easy
What does it mean to 'tile' a surface?
Example 4
easy
Can squares tile a flat surface by themselves?
Example 5
easy
Can regular pentagons tile a flat surface by themselves?
Example 6
easy
Name the three regular polygons that can tile a plane all by themselves.
Example 7
easy
How many squares meet at a single corner point in a square tiling?
Example 8
easy
How many equilateral triangles meet at a corner in a triangle tiling?
Example 9
easy
Can circles tile a flat surface with no gaps?
Example 10
easy
Why do honeycombs use hexagons?
Example 11
medium
Why must the angles meeting at a tiling vertex add up to exactly 360°?
Example 12
medium
A regular hexagon has interior angle 120°. How many meet at a tiling vertex?
Example 13
medium
Can a regular octagon (interior angle 135°) tile the plane alone?
Example 14
medium
Octagons and squares can tile together (like a stop-sign pattern). What is special about combining shapes?
Example 15
medium
Does every triangle (not just equilateral) tile the plane?
Example 16
medium
What is the difference between tiling and packing?
Example 17
medium
A regular 12-gon has interior angle 150°. Can it tile alone?
Example 18
medium
Why does the tiling test reduce to checking whether interior angle360° is a whole number?
Example 19
challenge
Prove that the only three regular polygons tiling the plane alone are the triangle, square, and hexagon.
Example 20
challenge
A floor is tiled with squares of side 4 inches. How many tiles cover a 96-inch by 72-inch floor?
Example 21
challenge
Why can any quadrilateral (even a non-convex one) tile the plane, while a regular pentagon cannot?
Example 22
challenge
What makes Penrose tilings remarkable compared to ordinary tilings?
Example 23
easy
A bathroom floor is covered in square tiles with no gaps and no overlaps. Is this a tiling?
Example 24
easy
Can you tile a flat surface using only circles?
Example 25
easy
Three regular polygons can tile a plane alone. Which one has 6 sides?
Example 26
easy
Two friends pave a patio using regular pentagons. Will it work?
Example 27
medium
A square tile costs $3. A floor needs 48 tiles. What is the total cost?
Example 28
medium
A rectangle floor is 300cm×180cm, tiled with 30cm square tiles. How many tiles?
Example 29
medium
A regular polygon has interior angle 144°. Can it tile the plane alone?
Example 30
hard
A regular polygon has 9 sides. Each corner has angle 140°. Does 360°÷140° give a whole number? Can it tile alone?
Example 31
hard
Tiles of two shapes — squares (90°) and triangles (60°) — meet at one corner. One arrangement is 4 triangles and 1 square. Does it fit?
Example 32
hard
Three different regular polygons meet at a corner. One is a triangle (60°), one is a square (90°). What is the third polygon's corner angle?
Example 33
challenge
Why does a Penrose tiling — using only two special shapes — never repeat its pattern?