A sliding equilateral triangle

Geometry Level 4

A unit equilateral triangle has two vertices sliding on the $x$ - and $y$ -axis, while the third vertex traces a curve, as shown above. The curve is an ellipse. Find its area.

Inspiration

\dfrac{\pi}{2}

\dfrac{\pi}{4}

\dfrac{2 \pi}{3}

\dfrac{\pi}{3}

\dfrac{3 \pi}{4}