ÿþ<html> <head> <title>MM of Everything: How to make a square from a circle.</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 make a square from a circle by construction"> <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 make a square from a circle.</FONT> </TD><TD> <img src="Greeknot.gif"> </TD></TR></TABLE> </H1> <P CLEAR=LEFT> <H4>In <a href="../MMoE.htm">The Murderous Maths of Everything</A> we meet <B>THE THREE UNSOLVED PROBLEMS OF ANTIQUITY</B>. <P>The ancient Greeks tried to do all their drawings just using a straight edge and a pair of compasses, but one of the things they found impossible was drawing a square that has exactly the same area as a circle. The Murderous Maths Organisation likes a challenge, so we solved the unsolved problem. <P> <TABLE border=1 bgcolor="004499"><TR><TD width=50% valign=top bgcolor="FFFFFF"> <font size=5><B>1/ </B></font> We need to start with a circle. If the diameter =1, then the circumference = <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span>. <BR clear=left> <img src="PI1.gif" > <BR clear=left> The formula for the area of any circle is <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span>d<sup>2</sup> / 4, so if d=1, the area of our circle = <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span>/4. <P> Our first job will be to make a rectangle with this same area. If we have a rectangle measuring <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span> along the bottom and 1/4 up the side then that'll do it! </TD><TD width=50% valign=top bgcolor="FFFFFF"> <img src="greek rules.gif" align=right> <font size=5><B>2/ </B></font> In the book we show how to convert the circumference of the circle into a straight line of length <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span>. That's the tricky part that the Greeks couldn't do - but we used a secret weapon! <BR clear=right> <img src="PI3.gif" > </TD></TR> <TR><TD bgcolor="FFFF44" colspan=2> <H4 clear=right>If the Greeks had managed to draw this diagram of the circle and the <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span> line, then they could have taken measurements off it and made a square with the same area as the circle. We didn't have space in the book to show exactly how they would have done it, so that's why this webpage is here! </TD></TR> <TR><TD width=50% valign=top bgcolor="FFFFFF"> <font size=5><B>3/ </B></font> To make our rectangle, we start by drawing a line of length <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span> and then construct a right angle at one end. <P> <img src="rightangle.gif" > </TD><TD width=50% valign=top bgcolor="FFFFFF"> <font size=5><B>4/ </B></font> Now we need to get a line of length 1/4. We get this by bisecting the diameter of the circle twice. (Remember the diameter = 1) Once we've done this we can use it to set our compasses to a length of 1/4. <P><img src="Diamquart.gif" > </TD></TR> <TR><TD valign=top bgcolor="FFFFFF"> <font size=5><B>5/ </B></font> Here's how to make the rectangle with area <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span>/4. We continue with our line that has the right angle constructed on the end. <P><img src="pirect.gif" > <P>This rectangle has the same area as the circle. </TD><TD valign=top bgcolor="FFFFFF"> <font size=5><B>6/ </B></font> So far so good. Now we need to convert this rectangle into a square of the same area <span lang=EN-GB style='font-family:Symbol; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"; mso-char-type:symbol;mso-symbol-font-family:Symbol'><span style='mso-char-type: symbol;mso-symbol-font-family:Symbol'>p</span></span>/4. This is a pretty good trick on its own! <P><img src="sqrect.gif" > </TD></TR></TABLE> <P>And that's how we solved one of the great UNSOLVED problems of antiquity using our SECRET WEAPON. It's such a brilliant weapon that we're surprised that the ancient Greeks didn't think of it first! <img src="trumps2.gif" align=right> <P> <P><BR> <img src="mmlogonew2.gif" ALIGN=LEFT> <FONT FACE = "ARIAL"> <P><a href="../MMoE.htm" > The Murderous Maths of Everything</a> <P><a href="../../indextxt.htm" > Murderous Maths Home Page</a> </body> </html>