Math Detours · Grade 5 · Around the corner · 7

Seven pieces, one area

The tangram's seven pieces make a square, a cat, a boat. The shape changes; the area cannot.

Name ____________________Date ____________

Tonight's idea

A tangram is a square cut into seven pieces: two large triangles, a medium triangle, two small triangles, a square and a parallelogram (a rectangle pushed over to one side).

4 across 4 up the same seven pieces
Left: the tangram on a grid. The two small triangles are orange. Right: all seven pieces again, moved and turned.

To make your own, draw a 4 by 4 square on squared paper, copy the lines, and cut. Lay all seven pieces flat, touching but never overlapping, and you can build a cat, a boat, and thousands more.

Area is how much flat space a shape covers. The big square is 4 by 4, so its area is 16 grid squares. Moving or turning a piece does not stretch it or shrink it. So every shape built from all seven pieces has area 16.

Rearranging the pieces changes the shape. It never changes the area.

Watch one: the area of a small triangle

Find the orange triangle on the right-hand edge of the square. It sits inside a rectangle 1 wide and 2 tall, and fills exactly half of it. The rectangle has area 2, so the small triangle has area 1.

Now the square piece. Cut it corner to corner and you get two small triangles. So the square piece has area \(1 + 1 = 2\).

Your turn

  1. Every piece can be cut into small triangles. How many small triangles make (a) the medium triangle, (b) the parallelogram, (c) one large triangle? Write down the area of each.
  2. Add up the areas of all seven pieces. Does your total match the 4 by 4 square?
  3. Take only the two small triangles. Join them along matching sides to make (a) a square, (b) a bigger triangle, (c) a parallelogram. Draw each one. What is the area of each?
  4. (a) Mia says she built a 5 by 5 square from all seven pieces, with no gaps. Explain why she must be wrong. (b) Leo builds a shape from every piece except the two small triangles. What is its area?

Stretch: how tall is the triangle?

The picture shows all seven pieces making one big triangle. Its bottom side is 8 long. A triangle is half of the rectangle drawn round it. How tall is the big triangle?

Answers

  1. (a) 2 small triangles, area 2. (b) 2 small triangles, area 2. (c) 4 small triangles, area 4.
  2. \(4 + 4 + 2 + 2 + 2 + 1 + 1 = 16\). Yes: the pieces fill the 4 by 4 square exactly.
  3. (a) Join the two long sides: a square. (b) Join two short sides so the other two short sides make a straight line: a triangle, the size of the medium one. (c) Join two short sides the other way round, with one triangle turned upside down: a parallelogram. Each has area \(1 + 1 = 2\). That is why three of the tangram pieces have the same area.
  4. (a) A 5 by 5 square has area 25. Seven pieces with total area 16 cannot cover 25 without gaps. (b) \(16 - 1 - 1 = 14\).

Stretch. The triangle has area 16, so the rectangle round it has area \(2 \times 16 = 32\). The rectangle is 8 long, and \(32 \div 8 = 4\). The triangle is 4 tall.

← PreviousNext: Shapes that tile →