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May 12th, 2014, 05:45 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, 06:40 AM  #2 
Senior Member Joined: Apr 2014 From: UK Posts: 895 Thanks: 328 
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, 08:09 AM  #3 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 155 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 $P+C=1$ $\implies$ $P=1C$. $\displaystyle \begin{align*} \therefore\ \frac{C^2}{4\pi}\ &=\ \frac{(1C)^2}{16} \\\\ 4C^2\ &=\ \pi(12C+C^2) \\\\ (4\pi)C^2+2\pi C\pi\ &=\ 0 \\\\ C\ &=\ \frac{2\pi+\sqrt{4\pi^24(4\pi)(\pi)}}{2(4\pi)} \\\\ &=\ \frac{2\sqrt{\pi}\pi}{4\pi} \\\\ &\approx\ 0.47 \end{align*}$ 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, 08:32 AM  #4 
Senior Member Joined: Apr 2014 From: UK Posts: 895 Thanks: 328 
Haha, got to laugh at my mistake there...


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area, circle, finding, similar, square 
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