The photograph shows the pieces of the insets—the decagon and the equivalent rectangle—and beneath each one there are the small equal triangles into which it can be subdivided. Here it is demonstrated that a rectangle equivalent to a decagon may have one side equal to the whole hypotenuse and the other equal to half of the perimeter.
Another inset shows the equivalence of the decagon and a rectangle which has one side equal to the perimeter of the decagon and the other equal to half of the altitude of each triangle composing the decagon. Small triangles divided horizontally in half can be fitted into this figure, with one of the upper triangles divided in half lengthwise.
Thus we demonstrate that the surface of a regular polygon may be found by multiplying the perimeter by half the hypotenuse.
SOME THEOREMS BASED ON EQUIVALENT FIGURES
A. All triangles having the same base and altitude are equal.
This is easily understood from the fact that the area of a triangle is found by multiplying the base by half the altitude; therefore triangles having the same base and the same altitude must be equal.
For the inductive demonstration of this theorem we have the following material: The rhombus and the equivalent rectangle are each divided into two triangles. The triangles of the rhombus are different, for they are divided by opposite diagonal lines. The three different triangles resulting from these divisions have the same base (this can be actually verified by measuring the bases of the different pieces) and fit into the same long rectangle which is found below the first three figures. Therefore, it is demonstrated that the three triangles have the same altitude. They are equivalent because each one is the half of an equivalent figure.
The decagon and the rectangle can be composed of the same triangular insets.
