# An Investigation of Historical Geometric Constructions - Calculating the Area of a Lune II

Author(s):
Suzanne Harper and Shannon Driskell

Since Heath noted that Hippocrates was initially “circumscribing a semicircle about an isosceles right-angled triangle”, it is necessary to begin by constructing an isosceles right-angled triangle as shown in figure 14.

Figure 14:  Isosceles right triangle ABC circumscribed by semicircle ABC.

An isosceles-right triangle could also have been constructed using points A, C, and D (see Figure 15). Therefore, AB = BC = CD = DA.

Figure 15: Construction of isosceles right triangle ABC.

Next, we construct a second circle with center D and radius DA to get “a segment of a circle similar to those cut off by the sides” of the isosceles right triangle  ABC (see Figure 16). This segment of a circle with base AEC is in a quadrant of a circle and is therefore similar to the two original segments with base AB and BC.

Figure 16:  Construction of a circle with center D.

Suzanne Harper and Shannon Driskell, "An Investigation of Historical Geometric Constructions - Calculating the Area of a Lune II," Convergence (August 2010)