<html> <head> <title>MM of Everything: How to turn a rectangle into a square.</title> <style><!--a:hover{color:FF2222; }--></style> </head> <body bgcolor="#99FFFF" text = "#4400AA" link="#008000" alink="#008000" vlink="#008000" > <meta name="keywords" content="Kjartan, Poskitt, Kjartan Poskitt, Murderous Maths, Murderous maths of everything, geometry, construction, greek, problem of antiquity,"> <meta name="description" content="How to turn a rectangle into a square of the same area."> <H1> <a href="../../indextxt.htm"> <img src="mmlogonew2.gif" align=LEFT hspace=20 border=0></A> <a href="../MMoE.htm"> <IMG SRC="MMoEcovsm.jpg" align=right border=0></A> <TABLE><TR><TD valign=top width=350><H1> <font color ="#ee0000">How to turn a rectangle into a square of the same area.</FONT> </TD><TD> <img src="happygreek.gif"> </TD></TR></TABLE> </H1> <P CLEAR=LEFT> <H4>In <a href="../MMoE.htm">The Murderous Maths of Everything</A> we meet the ancient Greek mathematicians. They liked to solve problems only using a pair of compasses and a straight edge. Some of the tricks they invented were ingenious, and this is one of our favourites! <P><center> <P><img src="sqrect.gif" border=2> </center> <P>Many thanks to MICHAEL JONES for supplying a neat little proof as to why this works! </H4> <P>Call the long side of the rectangle <b>a</b> and the short side <b>b</b>. The sides of the square are <b>s</b>. For the areas of the rectangle and square to be the same then <b>ab</b> must equal <b>s<sup>2</sup></b>. <P><center> <P><img src="sqrectproof.gif" border=2 align=right> </center> <P>If you make a little right angled triangle as shown, the hypotenuse (the longest side) = <b>(a+b)/2</b> and one of the short sides is <b>(a-b)/2</b>. The other short side is <b>s</b>. <P>Pythagoras Theorem says that <b>s<sup>2</sup> = [(a+b)/2]<sup>2</sup> - [(a-b)/2]<sup>2</sup> </b> <P>If we bang this through... <P><b>s<sup>2</sup> = (a<sup>2</sup> +2ab + b<sup>2</sup>)/4 - (a<sup>2</sup> -2ab + b<sup>2</sup>)/4 <P>s<sup>2</sup> = (a<sup>2</sup> +2ab + b<sup>2</sup> - a<sup>2</sup> + 2ab - b<sup>2</sup>)/4 <P>s<sup>2</sup> = 4ab/4 <P>s<sup>2</sup> = ab </b> <P> ...and that's what we wanted! <P><BR><a href="../BKMM7.htm" > <img src="../pix/mm7covsm.gif" ALIGN=RIGHT></A> <img src="mmlogonew2.gif" ALIGN=LEFT> <FONT FACE = "ARIAL"> <P><a href="../MMoE.htm" > The Murderous Maths of Everything</a> <P><a href="../BKMM7.htm" > Vicious Circles and other Savage Shapes</a> <P><a href="../../indextxt.htm" > Murderous Maths Home Page</a> </body> </html>