Constructions and Tilings · Lesson 8 of 12
Tiling with Tangrams
“Rearrange seven pieces into new outlines and learn what it means to cover a region without gaps or overlaps.”
• Identify all seven shapes in a tangram set. • Rearrange pieces without changing their combined area. • Use large pieces and boundary clues to approach a puzzle. • Distinguish a valid covering from gaps or overlaps. • Explain the meaning of tiling.
The Seven Tangram Pieces
Tangram puzzles originated in China and use seven pieces cut from a square. The pieces can be rearranged into a surprising variety of outlines, including letters, animals, and objects. The challenge is to match the target using the pieces themselves, rather than drawing a new shape.
There are two large triangles, one medium triangle, two small triangles, one square, and one parallelogram. A parallelogram is a four-sided figure with both pairs of opposite sides parallel. The triangle pieces are right isosceles triangles: each has one right angle and two equal shorter sides.
Cut a careful paper or cardboard copy of the labelled square into its seven pieces. Keep the cuts straight and do not trim pieces to make a puzzle fit. If you rotate or flip a piece, its lengths and area stay unchanged. The parallelogram may need flipping for some arrangements, depending on the puzzle and how the pieces are allowed to be used.
| Piece labels | Shape | Area compared with one small triangle |
|---|---|---|
| A and B | Large triangles | 4 each |
| C and E | Small triangles | 1 each |
| D | Square | 2 |
| F | Medium triangle | 2 |
| G | Parallelogram | 2 |
The table can be checked by comparing pieces or splitting them along equal triangle units. The two large triangles contribute eight small-triangle units in total. The square, medium triangle, and parallelogram contribute six more, and the two small triangles contribute two. The whole set therefore has sixteen such units.
Rearranging the Pieces
Every complete tangram arrangement uses the same material. The boundary changes because the pieces move, but the total area does not. This distinction helps you judge a proposed solution: a different outline is allowed; losing a piece or hiding it behind another is not.
Problem
The small triangle has area 3 square units. Find the area of the complete seven-piece set.
- 1.Each large triangle is four small units, so the two contribute 8 × 3 = 24 square units.
- 2.The square, medium triangle, and parallelogram are each two small units, contributing 6 × 3 = 18 square units together.
- 3.The two small triangles contribute 2 × 3 = 6 square units.
- 4.The whole set has area 24 + 18 + 6 = 48 square units. Every valid complete reassembly has this same area.
Forming an Arrow
An arrow is a useful first target because its tip and straight shaft suggest different jobs for the pieces. Look for the pointed head before filling the rectangular-looking shaft. When a source diagram shows internal piece boundaries, use them as a guide to the arrangement.
Begin by placing the pieces that form the arrowhead. Match their sloping edges at the tip and check where the head joins the shaft. Then use the remaining pieces to fill the shaft, matching straight edges. If the shaft is too wide or too narrow, reconsider the arrangement; do not resize a piece.
A useful puzzle strategy is to place the largest pieces early because they cover substantial regions and have fewer possible locations. Next inspect sharp corners and long straight edges. Leave smaller pieces until you can see which gaps need their particular angles. When two pieces touch, their shared edge must not leave a narrow uncovered strip.
Problem
A completed-looking target uses both large triangles, the medium triangle, the square, and one small triangle. Which pieces are missing?
- 1.Count the area units used: 4 + 4 + 2 + 2 + 1 = 13 small-triangle units.
- 2.The full set has 16 units, so 3 units of area are missing.
- 3.The unused pieces are the parallelogram, with 2 units, and the other small triangle, with 1 unit.
- 4.Area identifies the shortfall, but their exact positions must still be found using the target geometry.
Figure it Out
The source provides ten targets. Some have disconnected-looking parts that touch only at a point; others have long edges or pronounced corners. Try every silhouette using all seven pieces, and inspect both the outer outline and any small spaces inside it.
The diagrams below preserve the source shapes. The letter targets offer straight-edge clues, while the animal and object targets require more attention to piece orientation. If an attempt becomes stuck, take a piece away and explain what boundary it was meant to create. Changing one purposeful placement is more useful than shuffling all pieces randomly.
What Is Tiling?
Tangrams introduce a broader geometric idea. We can cover a specified region by fitting shapes together. A correct covering leaves no empty part of that region and places no piece interior on top of another piece interior.
Covering a region using a set of shapes without gaps or overlaps. Pieces may meet along their boundaries.
The seven-piece requirement belongs to this tangram task. Other tiling problems may allow repeated copies of one tile or a different collection of shapes. Always read which tiles are available and which moves are allowed. A piece placed outside the target does not cover a missing part inside it.
Problem
A seven-piece arrangement matches the target boundary, but one triangle partly covers the square. Is it a valid tiling?
- 1.Matching the outer boundary is not the only condition. The interiors must not overlap.
- 2.Where the triangle covers the square, two pieces contribute material to the same part of the region.
- 3.If all seven pieces have the target’s total area, that overlap means some other part must remain uncovered or some material must lie outside.
- 4.Move the pieces so their interiors are disjoint and the whole target is covered. The shown arrangement is not yet valid.
Area equality is a useful check, but it is not a complete puzzle solution. The pieces must fit the actual boundary and fill every part without overlap. Do not treat a correct total area as proof that the arrangement works.
Check Your Understanding
Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.
Quiz
How many pieces are in the standard tangram set?
Which shape list describes the set?
What happens to the total area when all pieces are rearranged without overlap?
Which arrangement is a valid tiling?
If the small triangle is one area unit, how many units are in the whole set?
Practice Problems
- Reassemble the original square from the seven pieces without looking at the labelled arrangement.
- Build the arrow shown in the lesson and identify at least three piece boundaries.
- Try all ten silhouette targets, beginning with the two letter-like targets.
- If a small triangle has area 2 square units, find the areas of the large triangle, square, and complete set.
- Explain how a target can have the correct total area yet still be covered incorrectly.
- Describe a systematic strategy for a tangram puzzle that you find difficult.
Use the two large triangles for the large diagonal regions, then fit the medium triangle, two small triangles, square, and parallelogram into the remaining area. All interiors must be disjoint.
Key Takeaways
• Tangrams use seven fixed pieces from a square. • The pieces include five triangles, a square, and a parallelogram. • Turning or flipping a piece preserves its lengths and area. • Every complete rearrangement has the same combined area. • Tiling means covering the region without gaps or overlaps. • Area checks support a solution but do not replace geometric fit.