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Inscribe Semicircle In Square by Geometric Construction

How would you inscribe a semicircle into a square using geometric construction? I prepared this slide show to demonstrate the steps of
construction. It can be viewed or downloaded at Semicircle In Square by Geometric ConstructionThe slide show and my geometry examples are Public Domain. There are some exercises on slide 8, with suggested solutions.
Why would you use this technique? It is not trivial to solve this problem
using algebra, as it involves factorisation of irrational numbers. Geometric Construction is an ancient
mathematical technique that can be done without knowing algebra. The basic tools are a
pair of compasses and a straight-edge or ruler. On a personal computer, you can use free geometry software such as Geogebra, as I did for these examples.

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Comment by Jaclyn on May 23, 2010 at 6:35am
I recently came across this posting and it reminded me of a problem I am currently working on. Maybe you can help because I'm stumped! Here goes: Farmer John stores grain in a large silo located at the edge of his farm. The cylinder-shaped silo has one flat, rectangular face that rests against the side of his barn. The height of the silo is 30 feet and the face resting against the barn is 10 feet wide. If the barn is approximately 5 feet from the center of the silo, determine the capacity of Farmer John's silo in cubic feet of grain.

Comment by Colin McAllister on May 26, 2010 at 3:33am
Jaclyn, the face resting against the barn is a chord of the circle, so you need an equation that relates chord length and the radius of the circle. You can derive this from the equation of a circle, x^2+y^2=r^2. It may help to visualise this if you draw the base of the silo on square graph paper. From the diagram, it is clear that x is half the chord length and y is the distance from the centre of the circle to the chord. This will give the radius of the circle. Then you can calculate the area of the missing part of the circle using integration with limits y=0 to y=r-5.
Comment by Colin McAllister on May 26, 2010 at 4:04am

It is evident from the diagram that the area of the missing part of the circle is 1/4 of the area of the circle, minus the area of two isosceles triangles (which add up to a square). Drawn using Geogebra.
Comment by Jaclyn on May 28, 2010 at 11:31pm
And this was a problem that was supposed to be an extension of exploring attributes of length, width, and volume and identifying and measuring attributes of a circle. I continue to find this problem far beyond the expectations of what the discussion pertained to for that week. I received my paper back with an unsatisfying grade, all because the volume was wrong. My instructor said the exact answer for the volume of the grain silo is 4282.5 cubic feet, but still fails to explain how to get the answer! Unbelievable. I appreciate YOU taking the time out to solve and illustrate the problem to me! I am certified to teach elementary grades K-5 and something tells me I will NEVER touch on this type of mathematics.
Comment by Colin McAllister on May 29, 2010 at 3:02am
The cylinder-shaped silo problem that ambushed Jaclyn would be clearer if it were expressed abstractly.
"A square is inscribed in a circle. Given that the square has side 10, (a) What is the radius of the circle? (b) What is the area of the circle? Consider the circle to be partitioned into 5 areas, one of which is the square itself. (c) What is the area of one of the segments that is bound by one side of the square and the circumference of the circle? (d) What area of the circle remains if one of these four segments is cut away?" (The question should reference a diagram of the situation.)
It would be sufficient to ask parts (a) and (b) of the question. Parts (c) and (d) are trivial arithmetic, but students that are capable of doing them would not get any marks, if they are blocked by parts (a) and (b).
Students need to be exposed to the language of abstract questions, before they are presented with them in an exam. (A surprising number of university students are not familiar with the word "given", as used to introduce the fixed properties of a scenario.)
I agree with Linda, that this problem is bad mathematics. There are two abhorrences in expressing the problem in farmyard language. (i) The description mystifies the problem rather than representing it, and it does so in a multitude of ways, one of which is to disguise a 2-D problem as 3-D. (ii) The problem is not one that a farmer or architect would need to solve.
A more realistic applied problem would be:
"A farmer has 1000 square feet of corrugated sheeting and wishes to use it to construct a grain silo against the flat wall of a very tall and wide barn. Recommend a design for the silo. Describe your design by drawing a floor plan and elevation, complete with relevant measurements or dimensions." Such a problem would be an advanced project with open-ended solutions, ideal for exercising teamwork and brain-storming, rather than an exam question, at any level.
Note: I am not a maths teacher, but I learned mathematics from one.
Comment by Colin McAllister on May 29, 2010 at 6:38pm

This picture shows my experiments with inscribed semicircles using the free Dr. Geo software. Using geometric construction, it is straight forward to generate recursively smaller inscribed semicircles, each half the area of the previous one. There are interesting coincidences in the diagram; the base of the 2nd semicircle is the side of the first square, and the corner of the third semicircle is the centre of the square.
Dr. Geo is interactive geometry software for the GTK desktop on Linux. (I had to edit the XML file of the square to set the corner points, to 0,0; 10,0; 10,10 and 0;10, as Dr. Geo doesn't support coordinate entry.) After exporting to PNG, I used Inkscape to colour the segments for clarity.
Comment by Colin McAllister on May 30, 2010 at 3:49pm

I noticed that figure of recursively inscribed semicircles contains a full circle inscribed in an isosceles right triangle. This can be inscribed more directly by drawing an arc from a corner of the triangle through a midpoint of the side. The opposite corner of the arc has a vertical that intersects the bisector of the triangle to define the centre of the circle.

The blue and red semicircles in the previous figure are self similar shapes, related by translation, rotation and scale. The construction is reproducible at any magnification or reduction that is a multiple of the square root of two. The self similarity resembles that of the Sierpinski triangle.
Comment by Colin McAllister on June 4, 2010 at 4:49am

I have updated my slideshow: Inscribe Semicircle In Square by Geometric Construction from 17 slides to 25 slides. This “Scorpion” was constructed by recursively adding isosceles right triangle, hypotenuse against short side. It contains a semicircle inscribed in a square.
Comment by Colin McAllister on April 4, 2011 at 2:57am
Linda Fahlberg-Stojanovska participated in this discussion, but unfortunately her valuable comments were deleted due to a technicality.


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