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How big of a square would I need to fit a 21'round pool in with an extra foot on all sides.I was 23' or 24'.I also thought the radius plus a foot. However it has been 6 years since my last college math course. I may be over thinking this.

Otis Buckner
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  • For clarification, 21' is the circumference of the pool? Or is it the radius or diameter of the pool? – JMoravitz Mar 16 '15 at 18:49
  • 21' circumference – Otis Buckner Mar 16 '15 at 22:52
  • Then, following my answer below, as it is a 21' circumference, that implies that the diameter of the pool will be $\frac{1}{\pi}\cdot 21$', so the sidelength of the circumscribed square is $\frac{1}{\pi}\cdot 21$', and you want to increase that by two. It will be approximately then $\approx 8.69'$. A pool with circumference 21' seems quite small however, as it will have diameter only 6.69' across, It would only be large enough for one adult male to lay down in, and not nearly large enough for anyone to swim in... more of a kiddy pool size. – JMoravitz Mar 16 '15 at 23:23

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There is a relationship between the diameter of an inscribed circle to the sidelength of the square in which the circle is inscribed. Namely, the diameter is equal to the sidelength.

If your measurements are not currently in terms of the diameter and the sidelengths of the circle and square respectively, then convert what measurements you do know into that information.

enter image description here

for the circle:

  • diameter = $2\cdot$ radius
  • diameter = $\frac{1}{\pi}\cdot$ circumference
  • diameter = $2\cdot \sqrt{\frac{\text{area}}{\pi}}$

for the square:

  • sidelength = $\frac{1}{4}\cdot$ perimeter
  • sidelength = $\sqrt{\text{area}}$

Once you have found the dimensions of the square which touches the edge of the circle (as pictured above), to allow for a foot of extra space on all sides, increase sidelength by 2 feet (since there needs to be an extra foot at the top and at the bottom, and left&right respectively).

note, the formulae used above come from $\text{area} = \pi\cdot r^2$, $d = 2\cdot r$, $c = 2\pi\cdot r$ etc...

If your pool is not circular, then you will need to use a more careful geometric approach related to your specificly shaped pool.

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JMoravitz
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