Inscribe Semicircle In Square by Geometric Construction


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Inscribe a Semicircle in a Square Using Geometric Construction. A slideshow showing the steps of construction. Public Domain. Exercises on slide 8.

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  • My latest presentation about paper-folding (origami) with interactive geometry software:
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  • The Geogebra and Dr Geo geometry data files for this presentation are on uploaded by colinmca.
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  • A follow up presentation about the geometry of Circles in Triangles:
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  • Linda Fahlberg-Stojanovska has published a paper on compass and straightedge construction, which includes a simpler proof and procedure for inscribing a semicircle in a square. At
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Inscribe Semicircle In Square by Geometric Construction

  1. 1. Inscribe a Semicircle in a Square Using Geometric Construction A slideshow showing the steps of construction
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  3. 3. Definition of Inscribed FigureIn geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. Specifically, at all points where figures meet, their edges must lie tangent. There must be no object similar to the inscribed object but larger and also enclosed by the outer figure. From Wikipedia, the free encyclopedia 3
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  9. 9. Exercise 1: Identify other largesemicircles that fit in a square or arectangle.Exercise 2: Draw circles and regularpolygons on paper using a pair ofcompasses and a ruler.Exercise 3: Find an alternativeprocedure for inscribing a semicirclein a square.Exercise 4: Draw large geometricshapes on a playground using chalk,string and straight lengths of wood.Exercise 5: Learn to use geometrysoftware on a personal computer. 9
  10. 10. Exercise 1.Identify other large semicircles that fit in a square. Some solutions follow: 10
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  14. 14. Exercise 3. Find an alternative procedure forinscribing a semicircle in a square. Hint: One method is to use the previous solution at half scale. 14
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  16. 16. Exercise 3. Alternative procedure for geometricconstruction of the inscribed semicircle. This solution was found by LindaFahlberg-Stojanovska, as described in her geometry paper at: The sequence of steps is numbered.
  17. 17. Calculation of Area ofSemicircleInscribed in Unit SquareThe unit square has a widthof 1. Divide this into two partsat the centre of the semicircle.Left of the origin, in the 9 oclockor negative-x direction,the distance to the side of thesquare is r, the radius.Right of the origin in the3 oclock or x directionthe distance is r/sqrt(2), bypythagorass theorem onthe 45 degree triangle.This expression can befactorised as follows tocalculate the radius r : 18
  18. 18. Recursively Inscribed SemicirclesUse compass and straightedge construction togenerate recursively smaller inscribed semicircles,each half the area of the previous semicircle. Theblue and red shapes are related by translation,rotation and scale. The construction is self similar,i.e. reproducible at any magnification or reductionthat is a multiple of the square root of two. Noticesome coincidences: The diameter of the secondsemicircle lies on the side of the first square, and thecorner of the third semicircle is the centre of thesquare. The figure also contains a circle inscribed inan isosceles right triangle.
  19. 19. Circles Inscribed in Isosceles Right TrianglesThere is a relationship between a circleinscribed in an isosceles right triangle, and asemicircle inscribed in a square. Beginningwith the brown triangle, we add a reducedgrey triangle. Notice that the centre of thecircle in the small triangle lies on the line thatdefines the blue square circumscribing thered semicircle in the large triangle.
  20. 20. ScorpionThis “Scorpion” is an extension of theprevious construction, by recursivelyadding isosceles right triangle,hypotenuse against short side. Theinscribed circles form a decreasingspiral, each one half the area of theprevious one. The construction wouldbe self similar, but for the largesttriangle, which is oriented 90 degreesoff of its proper position in the spiral.
  21. 21. Circles on CirclesA [brown] square is inscribed diagonally in a [grey]square, four [grey and yellow] isosceles right trianglesremain. When [white] circles are inscribed in thosetriangles, their centres lie on the circumference of the[yellow] circle that inscribes the outer square. Whenfour [brown] quarter circles are inscribed in the innersquare, corner on corner, a centred [white] circle,inscribed in the [blue] gap between them, is of thesame size as those inscribing the isosceles righttriangles. These coincidences arise from the ratio oflength of the diagonal to the side of a square, which isthe square root of two.
  22. 22. Circles on Circles ExtendedThe construction can be extended bycircumscribing another circle and squarearound the outer diagonal square.This series of constructions began bystudying a semicircle inscribed in a square. Ifyou look carefully, you can see semicirclesinscribed in squares, and circles inscribed inquarter circles.A suggested exercise is to inscribe fourcircles in a circle, using compass andstraightedge, or equivalent PC software.
  23. 23. This slide show and included geometricConstructions are in the public domain.Drawn using Geogebra and Dr. GeoOpen source geometry software.Radius and area of inscribed semicirclecalculated on the mathematics24x7.ning.comproblem solving group in collaboration withChristian, Steve and Danny. Alternativemethod of inscription discovered by Linda.Slideshow and constructions by Colin. 30