Question 1128669
A good strategy to calculate the percentages required is to
assign a length of {{{1}}} to a side or a radius,
because "scaling up or down" the drawing for any of the cases
would not change the ratio of areas of the geometrical figures.
 
a) The largest square inside a circle
{{{drawing(300,300,-1.1,1.1,-1.1,1.1,
circle(0,0,1),
green(triangle(-0.707,0.707,-0.707,-0.707,0.707,0.707)),
green(triangle(0.707,0.707,0.707,-0.707,-0.707,0.707)),
rectangle(-0.707,-0.707,0.707,0.707)
)}}}
If the radius of that circle is {{{r=1}}} ,
the area of the circle is {{{pi*r^2=pi*1^2=pi}}} ;
each of the {{{4}}} isosceles right triangles forming the square
has legs measuring {{{r=1}}} and area ={{{1*1/2=1/2}}} ,
and the area of the square is {{{2}}} .
The area of the square as a percentage of the area of the square as a fraction/percentage of the area of the circle is
{{{2/pi}}}{{{"="}}}{{{aproximately}}}{{{0.637=highlight("63.7%")}}}
 
b) The largest circle inside a square
{{{drawing(300,300,-1.1,1.1,-1.1,1.1,
circle(0,0,1),
rectangle(-1,-1,1,1)
)}}}
If the radius of that circle is {{{r=1}}} ,
the area of the circle is {{{pi*r^2=pi*1^2=pi}}} ;
the length of the side of the square is {{{2}}},
and the area of the square is {{{2*2=4}}} .
The area of the circle as a fraction/percentage of the area of the square is
{{{pi/4}}}{{{"="}}}{{{aproximately}}}{{{0.785=highlight("78.5%")}}}
 
c) The largest square inside a right isosceles triangle
{{{drawing(300,300,-0.1,2.1,-0.1,2.1,
green(triangle(0,0,1,0,0,1)),
triangle(0,0,2,0,0,2),
rectangle(0,0,1,1)
)}}}
There are 4 small congruent triangles inside the large right isosceles triangle,
with {{{2}}} of those small triangles forming the square,
so the square is {{{2/4=1/2=highlight("50.0%")}}} of the triangle.
 
d) The largest circle inside a right isosceles triangle
{{{drawing(300,300,-1.1,2.5,-1.1,2.58,
circle(0,0,1),
green(triangle(0,0,0,-1,-1,-1)),
green(triangle(0,0,-1,0,-1,2.414)),
green(triangle(0,0,2.414,-1,0.707,0.707)),
red(triangle(-1,-1,2.414,-1,0,0)),
red(triangle(-1,-1,-1,2.414,0,0)),
triangle(-1,-1,-1,2.414,2.414,-1)
)}}}
If the radius of that circle is {{{r=1}}} ,
and the length of each leg of the right isosceles triangle is {{{s}}} ,
the area of the circle is {{{pi*r^2=pi*1^2=pi}}} ,
and the area of the triangle is {{{s*s/2=s^2/2}}} .
The right isosceles triangle is made of three smaller triangles,
each with height {{{r=1}}} ,
and each having for a base one side of the right isosceles triangle.
Two of those triangles have a base of {{{s}}} ,
and the area of each one of those is {{{s*r/2=s/2}}} .
The third small triangle has {{{sqrt(2)s}}} for a base,
and its area is {{{sqrt(2)s*r/2=sqrt(2)s/2}}} .
The areas of those three small triangles adds up to the area of the right isosceles triangle, so
{{{s^2/2=s/2+s/2+sqrt(2)s/2}}} .
multiplying both sides of the equal sign times 2,
{{{s^2=s+s+sqrt(2)s}}} or {{{s^2=2s+sqrt(2)s}}} .
Dividing both sides of the equal sign by {{{s}}} ,
{{{s=2+sqrt(2)}}} .
So, the area of the right isosceles triangle is
{{{s^2/2=(2+sqrt(2))^2/2=(4+2+4sqrt(2))/2=(6+4sqrt(2))/2=3+2sqrt(2)}}} .
The area of the circle as a fraction/percentage of the area of the right isosceles triangle is
{{{pi/(3+2sqrt(2))}}}{{{"="}}}{{{aproximately}}}{{{0.539=highlight("53.9%")}}}