Material: Showing that a triangle is equal to a rectangle which has one side equal to the base of the triangle, the other side equal to half of the altitude of the triangle.
In a large rectangular metal frame there are two white openings: the triangle and the equivalent rectangle. The pieces which compose the rectangle are such that they may fit into the openings of either the rectangle or the triangle. This demonstrates that the rectangle and the triangle are equivalent. The triangular space is filled by two pieces formed by a horizontal line drawn through the triangle parallel to the base and crossing at half the altitude. Taking the two pieces out and putting them one on top of the other the identity of the height may be verified.
Already the work with the beads and the squaring of numbers has led to finding the area of a square by multiplying one side by the other; and in like manner the area of a rectangle is found by multiplying the base by half other. Since a triangle may be reduced to a rectangle, it is easy to find its area by multiplying the base by half the height.
Material: Showing that a rhombus is equal to a rectangle which has one side equal to one side of the rhombus and the other equal to the height of the rhombus.
The frame contains a rhombus divided by a diagonal line into two triangles and a rectangle filled with pieces which can be put into the rhombus when the triangles have been removed, and will fill it completely. In the material there are also an entire rhombus and an entire rectangle. If they are placed one on top of the other they will be found to have the same height. As the equivalence of the two figures is demonstrated by these pieces of the rectangle which may be used to fill in the two figures, it is easily seen that the area of a rhombus is found by multiplying the side or base by the height.
Material: To show the equivalence of a trapezoid and a rectangle having one side equal to the sum of the two bases and the other equal to half the height.
The child himself can make the other comparison: that is, a trapezoid equals a rectangle having one side equal to the height and the other equal to one-half the sum of the bases. For the latter it is only necessary to cut the long rectangle in half and superimpose the two halves.