document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=969785>Question 969785</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Each side of the squares equals to 7units.\r<BR>\n" );
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document.write( "How do I calculate the area of an equilateral triangle inside the square with sides 7units?\r<BR>\n" );
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document.write( "How do I work out the area of a circle inside a square with sides equal to 7 units?\r<BR>\n" );
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document.write( "What would the ratio be of area of triangle:area of circle? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #592681 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=592681\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> EQUILATERAL TRIANGLE:<BR>\n" );
document.write( "The area of a triangle with sides of lengths <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "opposite angles of measures <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C ALIGN=MIDDLE  ALT=\"C\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "can be calculated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=b%2Ac%2Asin%28A%29 ALIGN=MIDDLE  ALT=\"area=b%2Ac%2Asin%28A%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C9%2C-1%2C9%2C%0D%0Agreen%28triangle%280%2C0%2C3%2C8%2C3%2C0%29%29%2C%0D%0Agreen%28rectangle%283.5%2C0.5%2C3%2C0%29%29%2C%0D%0Atriangle%280%2C0%2C3%2C8%2C8%2C0%29%2Clocate%283.1%2C3.5%2Cgreen%28h=c%2Asin%28A%29%29%29%2C%0D%0Alocate%283.85%2C0%2Cb%29%2Carrow%283.8%2C-0.25%2C0%2C-0.25%29%2Carrow%284.2%2C-0.25%2C8%2C-0.25%29%2C%0D%0Alocate%285.5%2C4.5%2Cc%29%2Clocate%281.25%2C4.5%2Ca%29%2Clocate%288.2%2C0.2%2CA%29%2C%0D%0Alocate%28-0.4%2C0.2%2CC%29%2Clocate%282.9%2C8.5%2CB%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In the case of an equilateral triangle,<BR>\n" );
document.write( "all sides have the same length, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and all angles measure <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60%5Eo ALIGN=MIDDLE  ALT=\"60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=b%5E2%2Asin%2860%5Eo%29=b%5E2sqrt%283%29%2F2=approximately ALIGN=MIDDLE  ALT=\"area=b%5E2%2Asin%2860%5Eo%29=b%5E2sqrt%283%29%2F2=approximately\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.866b%5E2 ALIGN=MIDDLE  ALT=\"0.866b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "If your equilateral triangle inside a square looks like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C11%2C-1%2C11%2C%0D%0Agreen%28rectangle%280%2C0%2C10%2C10%29%29%2C%0D%0Ared%28triangle%280%2C0%2C10%2C0%2C5%2C8.66%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=s=7 ALIGN=MIDDLE  ALT=\"b=s=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=s%5E2%2Asin%2860%5Eo%29=%28sqrt%283%29%2F2%29%2As%5E2=approximately ALIGN=MIDDLE  ALT=\"area=s%5E2%2Asin%2860%5Eo%29=%28sqrt%283%29%2F2%29%2As%5E2=approximately\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.866%2As%5E2 ALIGN=MIDDLE  ALT=\"0.866%2As%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=7%5E2%2Asqrt%283%29%2F2=49sqrt%283%29%2F2=approximately ALIGN=MIDDLE  ALT=\"area=7%5E2%2Asqrt%283%29%2F2=49sqrt%283%29%2F2=approximately\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.866%2A49=42.434 ALIGN=MIDDLE  ALT=\"0.866%2A49=42.434\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "However, it may not be the expected answer,<BR>\n" );
document.write( "but you may be able to fit a slightly larger triangle if you \"tilt\" it,<BR>\n" );
document.write( "like this <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C11%2C-1%2C11%2C%0D%0Agreen%28rectangle%280%2C0%2C10%2C10%29%29%2C%0D%0Ared%28triangle%280%2C0%2C10%2C2.68%2C2.68%2C10%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "CIRCLE:<BR>\n" );
document.write( "The larges circle that you would be able to fit inside a square has a diameter as long as the side of the square:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C11%2C-1%2C11%2C%0D%0Agreen%28rectangle%280%2C0%2C10%2C10%29%29%2C%0D%0Ared%28circle%285%2C5%2C5%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > If the square side length is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s ALIGN=MIDDLE  ALT=\"s\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the circle radius is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=s%3F2 ALIGN=MIDDLE  ALT=\"r=s%3F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The area of a circle of radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%7D%7D+is+%7B%7B%7Bpi%2Ar%5E2 ALIGN=MIDDLE  ALT=\"r%7D%7D+is+%7B%7B%7Bpi%2Ar%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so a circle of radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%2F2%7D%7D+would+have+an+area+of%0D%0A%7B%7B%7Barea=pi%2A%28s%2F2%29%5E2=pi%2As%5E2%2F4 ALIGN=MIDDLE  ALT=\"s%2F2%7D%7D+would+have+an+area+of%0D%0A%7B%7B%7Barea=pi%2A%28s%2F2%29%5E2=pi%2As%5E2%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In particular, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=7 ALIGN=MIDDLE  ALT=\"s=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the area would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=pi%2A7%5E2%2F4=approximately ALIGN=MIDDLE  ALT=\"area=pi%2A7%5E2%2F4=approximately\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=38.485 ALIGN=MIDDLE  ALT=\"38.485\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "For the triangle and circle described above,<BR>\n" );
document.write( "the ratio of their areas would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%283%29%2F2%29%2As%5E2%2F%28pi%2As%5E2%2F4%29=%28sqrt%283%29%2F2%29%2As%5E2%2A%284%2Fpi%2As%5E2%29=sqrt%283%29%2A4%2As%5E2%2F%282pi%2As%5E2%29=2sqrt%283%29%2Fpi=approximately ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.103 ALIGN=MIDDLE  ALT=\"1.103\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "(The side of the square <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=7 ALIGN=MIDDLE  ALT=\"s=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > did not matter, as long as we use the same square size to fit the triangle and the circle).<BR>\n" );
document.write( " <BR>\n" );
document.write( "LARGER TRIANGLE:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-1%2C11%2C-1%2C11%2C%0D%0Agreen%28rectangle%280%2C0%2C10%2C10%29%29%2C%0D%0Ared%28triangle%280%2C0%2C10%2C2.68%2C2.68%2C10%29%29%2C%0D%0Alocate%280.1%2C5%2Cs%29%2Clocate%285%2C0%2Cs%29%2C%0D%0Alocate%286.3%2C10%2Cx%29%2Clocate%289.5%2C6.3%2Cx%29%2C%0D%0Alocate%280.5%2C10%2Cs-x%29%2Clocate%288.5%2C1.2%2Cs-x%29%2C%0D%0Alocate%285%2C1.9%2Cb%29%2Clocate%286%2C6.34%2Cb%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > In the right triangle corners, according to the Pythagorean theorem,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=x%5E2%2Bx%5E2 ALIGN=MIDDLE  ALT=\"b%5E2=x%5E2%2Bx%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=%28s-x%29%5E2%2Bs%5E2 ALIGN=MIDDLE  ALT=\"b%5E2=%28s-x%29%5E2%2Bs%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%5E2=%28s-x%29%5E2%2Bs%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%5E2=%28s-x%29%5E2%2Bs%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Bx%5E2=s%5E2-2sx%2Bx%5E2%2Bs%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2Bx%5E2=s%5E2-2sx%2Bx%5E2%2Bs%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=-2sx%2B2s%5E2 ALIGN=MIDDLE  ALT=\"x%5E2=-2sx%2B2s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2sx=2s%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B2sx=2s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B2sx%2Bs%5E2=2s%5E2%2Bs%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B2sx%2Bs%5E2=2s%5E2%2Bs%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Bs%29%5E2=3s%5E2 ALIGN=MIDDLE  ALT=\"%28x%2Bs%29%5E2=3s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bs%3E0 ALIGN=MIDDLE  ALT=\"x%2Bs%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%3E0 ALIGN=MIDDLE  ALT=\"s%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bs=sqrt%283s%5E2%29 ALIGN=MIDDLE  ALT=\"x%2Bs=sqrt%283s%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Bs=s%2Asqrt%283%29 ALIGN=MIDDLE  ALT=\"x%2Bs=s%2Asqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=s%281-sqrt%283%29%29 ALIGN=MIDDLE  ALT=\"x=s%281-sqrt%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=s%5E2%281-sqrt%283%29%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2=s%5E2%281-sqrt%283%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now, we knew that for a triangle, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%28sqrt%283%29%2F2%29b%5E2 ALIGN=MIDDLE  ALT=\"area=%28sqrt%283%29%2F2%29b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that this tilted triangle has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=x%5E2%2Bx%5E2=2x%5E2 ALIGN=MIDDLE  ALT=\"b%5E2=x%5E2%2Bx%5E2=2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=2%281-sqrt%283%29%29%5E2%2As%5E2 ALIGN=MIDDLE  ALT=\"b%5E2=2%281-sqrt%283%29%29%5E2%2As%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=area=%28sqrt%283%29%2F2%292%281-sqrt%283%29%29%5E2%2As%5E2=sqrt%283%29%281-sqrt%283%29%29%5E2%2As%5E2=approximately ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.928s%5E2 ALIGN=MIDDLE  ALT=\"0.928s%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That makes the area of thte tilted triangle<BR>\n" );
document.write( "approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.928%2A7%5E2=45.472 ALIGN=MIDDLE  ALT=\"0.928%2A7%5E2=45.472\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the ratio of areas for the tilted triangle and circle would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%283%29%281-sqrt%283%29%29%5E2%2As%5E2%2F%28pi%2As%5E2%2F4%29=sqrt%283%29%281-sqrt%283%29%29%5E2%2As%5E2%2A%284%2Fpi%2As%5E2%29=4sqrt%283%29%281-sqrt%283%29%29%5E2%2Fpi ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That ratio is approximately <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.182 ALIGN=MIDDLE  ALT=\"1.182\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );