Consider a thin uniform square sheet made of a rigid material. If its side is a, its mass m and its moment of inertia about one of its diagonals is I, then

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(3) I=(ma²)/(12). About the axis perpendicular to the sheet through its centre, a square plate of side a has I_z=m/(12)(a²+a²)=(ma²)/6. The two diagonals are in-plane axes through the centre, perpendicular to each other. Perpendicular-axis theorem: I_d₁+I_d₂=I_z. The square is symmetric under a quarter turn, so the two diagonals give the same value: I_d₁=I_d₂=I. 2I=(ma²)/6, so I=(ma²)/(12). This is the same value as about an axis through the centre parallel to a side — for a square, every in-plane axis through the centre gives the same moment of inertia.

Consider a thin uniform square sheet made of a rigid material. If its side is a, its mass m and its moment of inertia about one of its diagonals is I, then

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