What's the area of this square if it sits in this sector?

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\(ABCD\) is a square lying symmetrically inside a sector of a circle whose radius is \(\quantity{1}{in.}\), the angle of the sector being \(60^\circ\). \(A\) and \(B\) lie on the straight boundaries, \(C\) and \(D\) on the curved boundary. Calculate:

1. the length of the arc \(CD\),
2. the area \(ABCD\),
3. the area of the minor segment of the circle cut off by the chord \(CD\).

[Answers to (i) and (iii) may be left in terms of \(\pi\).]