1
$\begingroup$

Looking for the smallest circle which will hold three 1x1 squares. What is its radius?

By visualisation, I think that this is the optimal arrangement, but I'm unable to find the radius. Any hints would be appreciated.

image http://www2.stetson.edu/~efriedma/squincir/3.gif

  • 2
    Related: http://www2.stetson.edu/~efriedma/squincir/ ; r = 5 √17 / 16 = 1.288+ – 2017-01-24
  • 1
    Take three of the vertices of the squares which meet the circle. The radius of the circle is the circumradius of the triangle they form. – 2017-01-24
  • 0
    I'm completely unfamiliar with the concept of circumradius and how to apply it in this circumstance – 2017-01-24
  • 0
    https://latex.artofproblemsolving.com/1/d/7/1d7cc49568f0bcc67ca49a02ce767e44fdfc235a.png I saw this formula but I don't see how we can find out the length of the sides of the triangle. – 2017-01-24
  • 0
    @WanyuTang: for the side lengths it is enough to apply the Pythagorean theorem. – 2017-01-24

1 Answers 1

2

enter image description here

The side lengths of the highlighted triangle are $1$, $\sqrt{2^2+\left(\frac{3}{2}\right)^2}$ and $\sqrt{2^2+\left(\frac{1}{2}\right)^2}$ by the Pythagorean theorem. Since its area is $1$, we have: $$ \color{red}{R} = \frac{abc}{4\Delta} = \frac{1}{4}\sqrt{1\cdot\frac{25}{4}\cdot\frac{17}{4} }=\color{red}{\frac{5}{16}\sqrt{17}}\approx 1.28847. $$