Inscribe Semicircle In Square by Geometric Construction

Inscribe a Semicircle in a SquareUsing Geometric Construction

A slideshow showing thesteps of construction

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Definition of Inscribed Figure

In geometry, an inscribed planar shape or solidis 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 beno object similar to the inscribed object but

larger and also enclosed by the outer figure.

From Wikipedia, the free encyclopediahttp://en.wikipedia.org/wiki/Inscribed_figure

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Exercise 1: Identify other large semicircles that fit in a square or a rectangle.

Exercise 2: Draw circles and regular polygons on paper using a pair of compasses and a ruler.

Exercise 3: Find an alternative procedure for inscribing a semicircle in a square.

Exercise 4: Draw large geometric shapes on a playground using chalk, string and straight lengths of wood.

Exercise 5: Learn to use geometry software on a personal computer.

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Exercise 1. Identify other large semicircles that

fit in a square.

Some solutions follow:

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Exercise 3. Find an alternative procedure for

inscribing a semicircle in a square.

Hint: One method is to use theprevious solution at half scale.

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Exercise 3.

Alternative procedure for geometric construction of the inscribed semicircle.

This solution was found by Linda Fahlberg-Stojanovska, as described in

her geometry paper at:

http://geogebrawiki.pbworks.com/ciit10

The sequence of steps is numbered.

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Calculation of Area of SemicircleInscribed in Unit Square

The unit square has a widthof 1. Divide this into two partsat the centre of the semicircle.Left of the origin, in the 9 o'clockor negative-x direction,the distance to the side of thesquare is r, the radius.Right of the origin in the3 o'clock or x directionthe distance is r/sqrt(2), bypythagoras's theorem onthe 45 degree triangle.This expression can befactorised as follows tocalculate the radius r :

Recursively Inscribed Semicircles

Use compass and straightedge construction to generate recursively smaller inscribed semicircles, each half the area of the previous semicircle. The blue and red shapes are related by translation, rotation and scale. The construction is self similar, i.e. reproducible at any magnification or reduction that is a multiple of the square root of two. Notice some coincidences: The diameter of the second semicircle lies on the side of the first square, and the corner of the third semicircle is the centre of the square. The figure also contains a circle inscribed in an isosceles right triangle.

Circles Inscribed in Isosceles Right Triangles

There is a relationship between a circle inscribed in an isosceles right triangle, and a semicircle inscribed in a square. Beginning with the brown triangle, we add a reduced grey triangle. Notice that the centre of the circle in the small triangle lies on the line that defines the blue square circumscribing the red semicircle in the large triangle.

Scorpion

This “Scorpion” is an extension of the previous construction, by recursively adding isosceles right triangle, hypotenuse against short side. The inscribed circles form a decreasing spiral, each one half the area of the previous one. The construction would be self similar, but for the largest triangle, which is oriented 90 degrees off of its proper position in the spiral.

Circles on Circles

A [brown] square is inscribed diagonally in a [grey] square, four [grey and yellow] isosceles right triangles remain. When [white] circles are inscribed in those triangles, their centres lie on the circumference of the [yellow] circle that inscribes the outer square. When four [brown] quarter circles are inscribed in the inner square, corner on corner, a centred [white] circle, inscribed in the [blue] gap between them, is of the same size as those inscribing the isosceles right triangles. These coincidences arise from the ratio of length of the diagonal to the side of a square, which is the square root of two.

Circles on Circles Extended

The construction can be extended by circumscribing another circle and square around the outer diagonal square.

This series of constructions began by studying a semicircle inscribed in a square. If you look carefully, you can see semicircles inscribed in squares, and circles inscribed in quarter circles.

A suggested exercise is to inscribe four circles in a circle, using compass and straightedge, or equivalent PC software.

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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.