Shaded fraction of a square

$\begin{array}{l} \beta = {45^ \circ }\\ \chi = \arctan \left( {\frac{a}{{\frac{a}{2}}}} \right) = \arctan \left( {\frac{{2a}}{a}} \right) = \arctan \left( 2 \right)\\ \alpha = {180^ \circ } - \beta - \arctan \left( 2 \right) = {180^ \circ } - {45^ \circ } - \arctan \left( 2 \right) = {135^ \circ } - \arctan \left( 2 \right)\\ \frac{a}{{\sin \alpha }} = \frac{b}{{\sin \beta }} \Rightarrow b = \frac{{a \times \sin \beta }}{{\sin \alpha }} = \frac{{a \times \sin \left( {{{45}^ \circ }} \right)}}{{\sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right)}}\\ \frac{a}{{\sin \alpha }} = \frac{c}{{\sin \chi }} \Rightarrow c = \frac{{a \times \sin \chi }}{{\sin \alpha }} = \frac{{a \times \sin \left( {\arctan \left( 2 \right)} \right)}}{{\sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right)}}\\ s = \frac{{a + b + c}}{2} = \frac{{a + \frac{{a \times \sin \left( {{{45}^ \circ }} \right)}}{{\sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right)}} + \frac{{a \times \sin \left( {\arctan \left( 2 \right)} \right)}}{{\sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right)}}}}{2}\\ s = \frac{{a \times \sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right) + a \times \sin \left( {{{45}^ \circ }} \right) + a \times \sin \left( {\arctan \left( 2 \right)} \right)}}{{2 \times \sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right)}}\\ s = \frac{{a \times \left( {\sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right) + \sin \left( {{{45}^ \circ }} \right) + \sin \left( {\arctan \left( 2 \right)} \right)} \right)}}{{2 \times \sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right)}} \sim a \times \frac{{2.55}}{{1.897}} \sim a \times 1.34422773\\ {S_\Delta } = \sqrt {s \times \left( {s - a} \right) \times \left( {s - b} \right) \times \left( {s - c} \right)} \\ {S_\Delta } \sim \sqrt {a \times 1.34422773 \times \left( {a \times 1.34422773 - a} \right) \times \left( {a \times 1.34422773 - \frac{{a \times \sin \left( {{{45}^ \circ }} \right)}}{{\sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right)}}} \right) \times \left( {a \times 1.34422773 - \frac{{a \times \sin \left( {\arctan \left( 2 \right)} \right)}}{{\sin \left( {{{135}^ \circ } - \arctan \left( 2 \right)} \right)}}} \right)} \\ {S_\Delta } \sim \sqrt {a \times 1.34422773 \times \left( {a \times \left( {1.34422773 - 1} \right)} \right) \times \left( {a \times \left( {1.34422773 - 0.7454} \right)} \right) \times \left( {a \times \left( {1.34422773 - 0.9428} \right)} \right)} \\ {S_\Delta } \sim \sqrt {{a^4} \times 1.34422773 \times 0.34422773 \times 0.59882773 \times 0.40142773} \\ {S_\Delta } \sim {a^2} \times 0.33351394...\\ {S_\Delta } \stackrel{\textstyle.}{=} \frac{1}{3} \times {a^2}\\ S = {a^2}\\ \frac{{{S_\Delta }}}{S} = \frac{{\frac{1}{3} \times {a^2}}}{{{a^2}}} = \frac{1}{3} \times {a^2} \times \frac{1}{{{a^2}}} = \frac{1}{3} \end{array}$

The blue triangle is similar to the triangle above it.

Since its base is twice as large, its height is also twice as large.

The sum of the two heights is equal to the side of the square $a$ . Divide that in a ration 1:2 and you get the height of the blue triangle to be $\frac23 a$ .

Area of the triangle is therefore $A=\frac12 \cdot \frac 23 a\cdot a=\frac{a^2}3$ .

The ratio is then $\frac{a^2}3:a^2=\frac13\approx\boxed{0.3333}$