### Circumscribing Semicircles with Triangles

August 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.

Geometerss are well acquainted with circles, and also

polygons inscribed within, and

circumscribed around, circles. At about 250

BC, the

Greek mathematician,

Archimedes, circumscribed a

hexagon and polygons up to 96 sides around a circle to get

good estimates of the

mathematical constant,

pi.

A side of a hexagon circumscribed around a circle of unit

diameter is 1/√3, so that its

perimeter is 6/√3, or 3.46. A hexagon inscribed in a circle has a perimeter of 3.0 (see figure). The value of pi, Archimedes reasoned, was in between these values. The value found for a circumscribed 96-sided polygon is 22/7, the

elementary school approximation of pi (3.1429).

| *Hexagons inscribed within, and circumscribed around, a circle.*
The circumscribed hexagon gives an approximate value for pi of 3.46, while the inscribed hexagon gives 3.0. The actual value of pi (3.1415926535...) is between these estimates.
(Via Wikimedia Commons.) |

I find

plane geometry problems enjoyable, since they are quite simply stated. Some of them are also amenable to a

computer solution, so they allow me to practice my (limited)

computer programming skill.

In a recent

arXiv posting, Jun Li of the

Jiangxi University of Science and Technology (Ganzhou, China) presented the problem of finding the

triangle of smallest

area that can be circumscribed around a

semicircle.[1] He did an

analytical solution of the problem under a particular

constraint, but his

paper got me interested in the general problem.

As can be seen in the figure, triangles circumscribed around a semicircle come in many sizes. You can circumscribe

right triangles, fat triangles, and tall triangles. The constraint in this case is that the base of the triangle aligns with the

base of the semicircle, and the two other sides are

tangent to the

curve of the semicircle.

In searching for the circumscribed triangle of minimum area, I found that it was easiest to specify the

angles of the

altitudes from the base of the unit semicircle, as shown in the figure. It was then time to extract the long dormant

information I had learned in my

high school geometry class. As an indication of how long ago that was, I attended my 50th high school

reunion last year.

Some important ideas of this computation are that the

slopes of

perpendicular lines are their

negative reciprocals, and

intersecting points **(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:

- Jun Li, "The Triangle of Smallest Area Which Circumscribes a Semicircle," arXiv, June 27, 2016.