My Math Forum Finding the similar area of a square and circle

 Geometry Geometry Math Forum

 May 12th, 2014, 04:45 AM #1 Newbie   Joined: May 2014 From: australia Posts: 3 Thanks: 0 Finding the similar area of a square and circle If a piece of 1m wire is cut into 2 pieces to make a make a square and a circle with the same area how would you find this out?
 May 12th, 2014, 05:40 AM #2 Senior Member   Joined: Apr 2014 From: UK Posts: 960 Thanks: 342 Area of a circle (lets all it Ac) is [pi]r^2 Circumference of a circle, c, is 2[pi]r Therefore the area of a circle, with respect to circumference is: Ac = 4[pi]^3 / c^2 We know that the area of a square (lets call it As) is x^2, where x is the length of a side Ac = As as per the question, so: 4[pi]^3 / c^2 = x^2 we also know that c + 4x = 1 meter Which can be arranged as c = 1 - 4x We now have 2 equations with 2 unknowns, just substitute the 'c' 4[pi]^3 / (1 - 4x)^2 = x^2 The numbers in this are horrible.... I'll leave it at that stage (tired!) But that's how you go about solving these types of question.
 May 12th, 2014, 07:09 AM #3 Senior Member     Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra For the circle with circumference $C$ and radius $r$: $C^2=4\pi^2r^2=4\pi A$ $\implies$ $A=\dfrac{C^2}{4\pi}$ For the square with perimeter $P$ and side $x$: $P^2=16x^2=16A$ $\implies$ $A=\dfrac{P^2}{16}$. And $P+C=1$ $\implies$ $P=1-C$. \displaystyle \begin{align*} \therefore\ \frac{C^2}{4\pi}\ &=\ \frac{(1-C)^2}{16} \\\\ 4C^2\ &=\ \pi(1-2C+C^2) \\\\ (4-\pi)C^2+2\pi C-\pi\ &=\ 0 \\\\ C\ &=\ \frac{-2\pi+\sqrt{4\pi^2-4(4-\pi)(-\pi)}}{2(4-\pi)} \\\\ &=\ \frac{2\sqrt{\pi}-\pi}{4-\pi} \\\\ &\approx\ 0.47 \end{align*} So the wire should be cut to pieces of lengths approximately $\mathrm{47\ cm}$ and $\mathrm{53\ cm}$, the former to make the circle and the latter to make the square.
 May 12th, 2014, 07:32 AM #4 Senior Member   Joined: Apr 2014 From: UK Posts: 960 Thanks: 342 Haha, got to laugh at my mistake there...

 Tags area, circle, finding, similar, square

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Raye Algebra 1 November 27th, 2013 01:42 PM Tutu Algebra 3 May 8th, 2012 07:14 AM zerostalk Algebra 6 January 10th, 2012 08:06 PM gus Algebra 1 April 17th, 2011 04:25 PM vnonwong Algebra 3 October 14th, 2009 05:09 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top