Circle incribed within a triangle percentage 
I have worked out the areas as $\pi/3$ for the circle and $2/\sqrt3$ for the triangle but don't know how to convert into a percentage without a calculator.
 A: Just replace $\pi$ with 3.14 and $\sqrt3{}$ with 1.73 (do the students today know how to do the square root by hand any longer?), and calculate approximately. Than see what percentage is closest.
A: If the radius of the circle is $1$ then the area of the circle is $\pi$ and the area of the triangle is $3\sqrt{3}$
Therefore the percentage is $$\frac{\pi}{3\sqrt{3}}\times100\simeq\frac{100}{\sqrt{3}}\simeq\sqrt{\frac{10000}{3}}\simeq\sqrt{3300}\simeq60%$$
A: $$
\begin{gathered}
  side_{\,tr}  = 1\quad \quad h_{\,tr}  = \frac{{\sqrt 3 }}
{2}\quad \quad r_{\,circ}  = \frac{{\sqrt 3 }}
{6} \hfill \\
  A_{\,circ}  = \pi \frac{3}
{{36}} = \frac{\pi }
{{12}}\quad \quad A_{\,tr}  = \frac{{\sqrt 3 }}
{4} \hfill \\
  \rho  = \frac{{A_{\,circ} }}
{{A_{\,tr} }} = \frac{\pi }
{{12}}\frac{4}
{{\sqrt 3 }} = \frac{\pi }
{3}\frac{1}
{{\sqrt 3 }} \approx \frac{{3.15}}
{3}\frac{1}
{{\sqrt 3 }} = \frac{{1.05}}
{{\sqrt 3 }} \hfill \\
  \rho ^{\,2}  \approx \frac{{\left( {1.05} \right)^{\,2} }}
{3} = \frac{1}
{3}\left( {1 + \frac{5}
{{100}}} \right)^{\,2}  \approx \frac{1}
{3}\left( {1 + 2\frac{5}
{{100}}} \right) = \frac{{110}}
{{300}} \approx \frac{{36}}
{{100}} \hfill \\
  \rho  \approx \sqrt {\frac{{36}}
{{100}}}  = \frac{6}
{{10}} = \frac{{60}}
{{100}} = 60\%  \hfill \\ 
\end{gathered} 
$$
