Dissected Square, Ross Surface Form #11

This is the eleventh in a series of models of plane figures (surface forms) designed by William Wallace Ross, a school superintendent and mathematics teacher in Fremont, Ohio. The unpainted wooden square is bisected along one diagonal, with wooden dowels to hold the pieces together. Along the other diagonal, one of the triangles is bisected. On the back, an inscribed circle is indicated as well as the radius of the circle and a square inscribed inside it. A paper label glued to the object reads: SQUARE 6x6.

This is the first in a series of models in which Ross considered the area of a regular polygon to be made up of isosceles triangles, with base equal to the length of the side of the polygon and height equal to the radius of the inscribed circle. Summing the area of the triangles, he found that the total area equaled half the perimeter of the polygon times the radius. In this case, each of the four triangles had base 6, height 3 (the radius of the inscribed circle), and area 9. The perimeter is 4x6 or 24, half the perimeter is 12, and the area is 36. This was the same area found by multiplying the length of the sides of the square.

Although they are not listed in his 1891 Manual, Ross also would make models of regular polygons with 8 and with 16 sides, similarly divided into triangular sectors. Examples of these have catalog numbers 1985.0112.201 and 1985.0112.202. He then, like A. H. Kennedy before him, generalized the dissection to represent the area of a circle (see 1985.0112.203).

For further information about Ross models, including references, see 1985.0112.190.

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