## Circumscribing Semicircles with TrianglesAugust 15, 2016 Humans are fascinated by circles, so much so that many objects are circular in shape, many expressions contain the word, circle, and many English words begin with the prefix, "circum-," derived from the Latin word for circle. One massive online word list has more than 250 words starting with circum, including circumlocution, circumspect, circumstantial, and circumvent.
(x,y) are found by equating the line equations, y = mx + b, where m is the slope and b is the y-intercept. The sine and cosine functions are useful for finding the length of the sides of right triangles when an angle and the length of another side are known.
In the past, a computer program of this sort would use a minimum value search algorithm akin to the method of steepest descent. Today's computers are so fast that a lazy programmer can do an exhaustive search in a problem like this, perhaps with a reduction in the variable increments as they get close to the goal; or, the Monte Carlo method that I used, in which you try random values and keep track of how well you're doing. My C language source code, in my usual amateur coding style, can be found here
So what's the result? A proper mathematician might have told me that it was obvious that an isosceles right triangle (θ_{1} = θ_{2} = 45°) gives the minimum area. For a semicircle radius of one, the area is 2.0. This can be seen from the contour plot, shown below, that also indicates how slowly the area changes with angle.
## Reference:
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