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Semicircle in Square

Page history last edited by LFS 10 years, 2 months ago

 Home > Construct with GeoGebra -> Advanced Contructions -> Inscribe Semicircle in Square

 Inscribe a Semicircle in Square

TOC - Advanced C+S Triangle from Medians 
 

Goal:  a semicircle in a square.  Idea by Colin McAllister

Inscribe: To draw one figure within another figure so that every vertex of the enclosed figure touches the outer figure (Free Dictionary). In 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 (Wikipedia).

YouTube Mathcast or ScreenCast Mathcast (if YT is blocked)
YT here
SC here
 
 
GeoGebra Exploration:  Given square, create an exploring constructionDirections for Exploration Interactivity (opens in new window)
 
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
 
 
GeoGebra Construction: Watch the C+S construction and then test it Directions for InterActivity
 

1. Click on Play to see the construction unfold.

2. Select the   and click & drag points A and/or B. The semi-circle is always inscribed.

 
Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
 
 
Proof (Colin McAllister)
 

An analysis of the two congruent triangles AGE and GJC is proof that we have identified the inscribed semicircle. The geometric construction can be used to calculate the radius of the semicircle inscribed in a square of side a. We have drawn a circle of radius=a centered on corner C. It intercepts the diagonal of length=a*sqrt(2) at a point Z, which is therefore a distance: a*sqrt(2)-a from corner A. By drawing a circle centered on A through Z, we identify a point H on side AB. The distance HB=AB-AH, so that HB=a-(a*sqrt(2)-a). Thus, HB=a*(2-sqrt(2)). The vertical line from H crosses the diagonal at a point G, which is the centre of the semicircle. So the radius r of the inscribed semicircle is r=a*(2-sqrt(2)). In the case of a unit square with side a=1, the radius is r=2-sqrt(2). We can also calculate the area of the inscribed semicircle, which is simply pi*r*r/2. This is the largest semicircle that fits in the square.

Note: sqrt() represents the square root function. Calculations can be represented exactly using the square root of 2, which is an irrational number, or approximately using a decimal representation of the square root of 2, to a chosen number of decimal places.

 
A triangle with medians ta+ tb and tcexists and is unique if and only if the 3 medians satisfy the triangle inequality.
 

 
 

Metadata

Global Advanced Compass and Straightedge Constructions with GeoGebra
Brief Construction: Inscribe a Semicircle in a Square
Grade 10th grade and up -Geometry
Strand Geometry
Standards CA Geometry 16.0
Keywords construction, straightedge, compass, ruler, geogebra, geometry, angle, triangle, medians, copy
Comments none
Download You can download everything; a zip is in progress ...
Author LFS - contact
Type Freeware - Available for Offline and Online Use - Translatable (html)
Use Requires sunJava player

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