Solution 1:
r=45^\circ
x^2 = 45^2 + 45^2 - 2 \cdot 45 \cdot 45 -\frac{\sqrt3}{2} = 2 \cdot 45^2 - 45^2\cdot \sqrt{3}
A = 12 \cdot \frac{1}{2} \cdot 45 \cdot 45 \cdot \frac{1}{2}
= 3 \cdot 45^2
A_\triangle = \frac{\sqrt{3}}{4} \cdot x^2 = \frac{\sqrt{3}}{4} (2 \cdot 45^2 - 45^2\cdot \sqrt{3})
8A_\triangle = 2\sqrt{3} (2 \cdot 45^2 - 45^2\cdot \sqrt{3})
4A_\square = 8 \cdot 45^2 - 4\cdot 45^2\sqrt{3}
8A_\triangle = 4\sqrt{3} \cdot 45^2 - 6 \cdot 45^2
4A_\square + 8A_\triangle = 2 \cdot 45^2
A = 45^2 = 2025
[Note: x
Solution 2:
(x + \frac{\sqrt{3}}{2}x)^2 + (\frac{x}{2})^2 = 45^2
x^2 + \frac{3}{4}x^2 + \sqrt{3}x^2 + \frac{x^2}{4} = 45^2
(2+\sqrt{3})x^2 = 45^2
x^2 = \frac{45^2}{2 + \sqrt{3}}
\frac{1}{2}x(\frac{\sqrt{3}}{2}x+x) = \frac{\sqrt{3} + 2}{4} x^2
A = 12 \cdot \frac{\sqrt{3} + 2}{4}x^2
= 3(\sqrt{3} + 2) \cdot \frac{45^2}{2 + \sqrt{3}}
= 3 \cdot 45^2
A_\triangle = \frac{\sqrt{3}}{4}x^2 \cdot 8
= \frac{\sqrt{3}}{4} \cdot \frac{45^2}{2 + \sqrt{3}} \cdot 8
= \frac{2\sqrt{3} \cdot 45^2}{2 + \sqrt{3}}
A_\square = x^2 \cdot 4
= \frac{4 \cdot 45^2}{2 + \sqrt{3}}
A_{\triangle + \square} = \frac{2\sqrt{3} \cdot 45^2 + 4 \cdot 45^2}{2 + \sqrt{3}}
= \frac{2 \cdot 45^2 \cdot (\sqrt{3} + 2)}{2 + \sqrt{3}}
= 2 \cdot 45^2
A_{shaded} = 3 \cdot 45^2 - 2\cdot 45^2
= 45^2
= 2023
[Note: x
