Tangram areas

Cut out the seven tangram pieces from the end of your textbook. Together they make one big square. The pieces are named A to G:

• A and B are the two big triangles.
• C and E are the two small triangles.
• D is the small square.
• F is a middle-sized triangle.
• G is a slanting four-sided piece.

How can we compare their areas without any numbers? Place one piece over another. If one piece covers the other exactly, with nothing sticking out and nothing left uncovered, the two pieces have the same area.

Doing this, the book finds that A and B have the same area, and C and E have the same area. The small square D cannot be covered by C alone, but it can be covered exactly by C and E put together. So D has the same area as C and E added together. Since C and E are equal, D has twice the area of C (or of E).

Now make a rectangle using all seven pieces. The shape changes, but the pieces are the same, with no gaps and no overlaps, so the rectangle has exactly the same area as the big square. Moving pieces around never changes the total area.