Geometry · Perimeter, area & similarity
Tangram Geometry Lab
Seven pieces, one square.
Seven flat pieces cut from one 10 cm square. You will measure them, compare them, and prove that nothing is lost when you take the square apart.
Download the printable worksheet PDF · 9 pages · 430 KB
Background
A short history of the tangram
The tangram is a dissection puzzle: a square cut into seven pieces called tans. The rules are simple. Use all seven pieces, lay them flat, and do not let them overlap. With those rules people have built thousands of silhouettes: animals, people, boats, letters and numbers.
- Song dynastyHistorians often trace the idea to yanjitu, sets of banquet tables described by the Song scholar Huang Bosi that could be pushed together into different shapes. A Ming-dynasty version used triangular "butterfly" tables. These were furniture, not puzzles, but they used the same idea of rearranging fixed parts.
- c. 1800The puzzle itself appears in China, called qīqiǎobǎn (七巧板), "seven boards of skill." The oldest surviving tangram books were printed in the early 1800s, around 1813.
- 1815–1820Trading ships carried tangram sets and books to America and Europe. A "tangram craze" swept Europe around 1817–1818, with puzzle books printed in English, French, German, Italian and Danish. Popular stories claim famous people of the time played it; many of these stories are hard to verify.
- Mid-1800sThe English word tangram appears. Nobody is sure where it comes from. One guess joins "Tang" (an old name for China) with "-gram" (something drawn or written).
- 1903American puzzle maker Sam Loyd published The Eighth Book of Tan, claiming the puzzle was 4,000 years old and invented by a god named Tan. It was a joke. Loyd made the whole history up, but the myth spread for decades.
- 1942Mathematicians Fu Traing Wang and Chuan-Chih Hsiung proved that exactly 13 convex shapes (shapes with no dents) can be made from the seven tans.
Today tangrams are used in classrooms around the world because every piece is built from the same right isosceles triangle. That makes them a perfect tool for studying area, perimeter, angles and similarity.
Meet the pieces
The seven tans, with measurements
All figures are drawn at the same scale, so you can compare their sizes directly. Lengths are rounded to two decimal places. Dashed lines show a height (altitude) or diagonal you may find useful. A small square marks a right angle.
Part A
Perimeter and area of each piece
Use the measurements above. Perimeter is the distance around a shape: add all the side lengths. For area, use these formulas:
- Triangle: A = ½ × base × height. In a right triangle, the two legs can be the base and height.
- Square: A = side × side
- Parallelogram: A = base × height (the height is perpendicular to the base, not the slanted side)
| Piece | Shape | Perimeter (cm) | Area (cm²) |
|---|---|---|---|
| A | Large triangle | ||
| B | Large triangle | ||
| C | Medium triangle | ||
| D | Small triangle | ||
| E | Small triangle | ||
| F | Square | ||
| G | Parallelogram |
Tip: if you square 3.54 you get 12.53, not 12.5, because 3.54 is rounded. Answers within 0.1 count as correct. Using the dashed heights avoids the rounding problem.
- A, B: P = 7.07 + 7.07 + 10 = 24.14 cm (exactly 10 + 10√2). Area = ½ × 10 × 5 = 25 cm².
- C: P = 5 + 5 + 7.07 = 17.07 cm (10 + 5√2). Area = ½ × 5 × 5 = 12.5 cm².
- D, E: P = 3.54 + 3.54 + 5 = 12.07 cm (5 + 5√2). Area = ½ × 5 × 2.5 = 6.25 cm².
- F: P = 4 × 3.54 = 14.14 cm (10√2). Area = 3.54² ≈ 12.5 cm² (exactly (5√2⁄2)² = 50⁄4 = 12.5, or ½ × diagonal² = ½ × 25).
- G: P = 5 + 3.54 + 5 + 3.54 = 17.07 cm (10 + 5√2). Area = 5 × 2.5 = 12.5 cm².
Part B
Congruent and similar figures
Congruent figures have exactly the same shape and size: one fits perfectly on top of the other. Similar figures have the same shape (equal angles, sides in the same ratio) but can be different sizes. Every pair of congruent figures is also similar.
Which pairs of pieces are congruent? Explain how you know.
Write your answer here, or on paper.A and B are congruent (legs 7.07, hypotenuse 10). D and E are congruent (legs 3.54, hypotenuse 5). These are the only congruent pairs.
Which pieces are similar to each other? List every group.
All five triangles (A, B, C, D, E) are similar. Each is a right isosceles triangle with angles 45°, 45°, 90°. The square F and the parallelogram G are not similar to any other piece.
Find the scale factor from a small triangle to the medium triangle, and from the medium triangle to a large triangle. (Divide matching sides, for example hypotenuse ÷ hypotenuse.)
Small → medium: 7.07 ÷ 5 ≈ 1.41 (exactly √2). Medium → large: 10 ÷ 7.07 ≈ 1.41 (√2). Small → large: 10 ÷ 5 = 2.
Compare the areas of the small, medium and large triangles. When the side lengths are multiplied by the scale factor k, what happens to the area?
Areas are 6.25, 12.5 and 25 cm², a ratio of 1 : 2 : 4. Each time the sides grow by √2, the area doubles, because the area is multiplied by k² = (√2)² = 2. From small to large, k = 2, so the area is multiplied by 2² = 4.
The medium triangle C, the square F and the parallelogram G all have the same area. Are any of them congruent? Are any of them similar? Explain.
No. They have equal area but different shapes and different angles (a triangle, a square and a parallelogram), so they are neither congruent nor similar. Equal area does not mean congruent.
Part C
The completed square
Which piece edges make up the top side of the square? Use them to find the side length.
cmFind the perimeter of the square.
cmFind the area of the square.
cm²
The top edge is the hypotenuse of large triangle A, so the side is 10 cm. (The left side is the hypotenuse of B. The right side is D's hypotenuse plus one leg of C: 5 + 5 = 10.) Perimeter = 4 × 10 = 40 cm. Area = 10 × 10 = 100 cm².
Part D
Do the pieces add up?
When you cut a shape into pieces, no area is created or lost. Show that this is true for the tangram: copy your areas from Part A, add them, and compare the total with the area of the square from Part C.
| Piece | Area (cm²) | Fraction of the square |
|---|---|---|
| A · Large triangle | ||
| B · Large triangle | ||
| C · Medium triangle | ||
| D · Small triangle | ||
| E · Small triangle | ||
| F · Square | ||
| G · Parallelogram | ||
| Total of the seven pieces |
Finish the statement: 25 + 25 + 12.5 + 6.25 + 6.25 + 12.5 + 12.5 = ____ cm² = area of the square.
25 + 25 + 12.5 + 6.25 + 6.25 + 12.5 + 12.5 = 100 cm², which equals the square's area of 10 × 10 = 100 cm².
Fractions: A and B are ¼ each; C, F and G are ⅛ each; D and E are 1⁄16 each. ¼ + ¼ + ⅛ + ⅛ + ⅛ + 1⁄16 + 1⁄16 = 16⁄16 = 1.
Part E · Go further
More things to notice
Pick at least two of these. Use a paper tangram if you can.
Area is kept, perimeter is not
Add up the perimeters of all seven pieces. Compare the total with the square's perimeter of 40 cm. Why is the total so much bigger?
Total ≈ 2(24.14) + 17.07 + 2(12.07) + 14.14 + 17.07 = 120.71 cm, about 3 times 40 cm. Inside edges are shared by two pieces and get counted twice, but they are not on the outside of the square.
The small triangle as a unit
How many small triangles would cover each piece? How many would cover the whole square?
Large = 4, medium = 2, square = 2, parallelogram = 2, small = 1. The whole square is 4 + 4 + 2 + 1 + 1 + 2 + 2 = 16 small triangles, matching 100 ÷ 6.25 = 16.
Same area, different shapes
Build a rectangle, a triangle and a parallelogram using all seven pieces. Each has an area of 100 cm². Do they have the same perimeter?
No. Shapes with the same area can have very different perimeters. A square has the smallest perimeter of any rectangle with a given area.
Same perimeter twins
Two different pieces have the same area and the same perimeter. Which ones?
The medium triangle C and the parallelogram G: both 12.5 cm² and 17.07 cm. They are still not congruent.
Angles in 45° steps
List every angle in all seven pieces. What do you notice? Why does this help the pieces fit together?
Every angle is 45°, 90° or 135°, all multiples of 45°. Angles that meet at a point must add to 360° (or 180° along a straight edge), and multiples of 45° combine easily to make those totals.
The piece you must flip
The parallelogram is the only piece that cannot match its own mirror image by sliding or turning. Try it. Some tangram pictures need it flipped over.
A parallelogram that is not a rectangle or rhombus has no line of symmetry, so its reflection can only be matched by flipping it over. Every other tan has a line of symmetry.
Where √2 hides
Several lengths are multiples of √2, which is irrational. Can a tangram cut from a 10 cm square have every side length be a whole number of centimetres?
No. In a right isosceles triangle the hypotenuse is always √2 times a leg, so the legs and the hypotenuse can never both be whole numbers. That is why the decimals never quite "come out even."
The 13 convex shapes
Wang and Hsiung proved only 13 convex shapes can be made from all seven pieces. How many can you find? (Hint: the square, a rectangle, a triangle and a trapezoid are among them.)
The 13 are: 1 triangle, 6 quadrilaterals, 2 pentagons and 4 hexagons.