The mathematics involved in drawing a circle using a carpenter's square.
Starting with Figure 7, lets label the vertices and to make things simpler, assume that vertex A is (0,0), hence vertex C is (D,0). See Figure 8.
Angle ABC is a right angle and its hypotenuse triangle is segment AC, which has fixed length D. If we know angle BAC, denoted by , then we can determine the length of side AB and the length of side BC. We use the following notation:
It follows that
Next we drop a perpendicular from vertex B to side AC as shown in Figure 9. Label the intersection of the perpendicular and side AC as E.
Triangle ABE is a right triangle and x = length of AE while y = length of BE. Since we can compute the length of AB and angle is taken as known, we can compute x and y as follows:
Using that we can substitute in the previous relations to get
This development is a nice exercise in right triangle trigonometry that is accessible at a variety of levels.
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