There are two squarable lunes that can be created from a 30-60-90 triangle. One lune is created off the short leg of the triangle and the other lune is created off of the long leg of the triangle. To construct these lunes we begin with our 30-60-90 triangle.

To construct both of these lunes be begin by circumscribing the 30-60-90 triangle within a circle whose diameter is the hypotenuse of our triangle.

We will first construct the lune that is off the short leg of the 30-60-90 triangle.

We begin by constructing a circle whose diameter is the short leg of the triangle.

The lune we are seeking is the one that is formed by the area of the small circle around the short leg that is not inside the area of our larger circle whose diameter is the hypotonuse. This lune is shaded blue.

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The second lune is constructed off the long leg of the 30-60-90
triangle.

We begin by constructing a circle whose diamter is the long leg of the triangle.

The lune we are seeking is formed by the area of the small circle around the long let that is not inside the area of our larger circle whose diameter is the hypotenuse. This lune is shaded blue.