Physical properties are important for the selection of concrete armour units (CAUs) for a specific site. Geometric properties are closely linked to physical properties. Here, new concepts in geometric properties that may be related to structural stability are proposed. Void ratio, overall slenderness, member slenderness, mass distribution with the distance from the gravity centre, and moment of inertia with respect to the gravity centre or pivot line are measurable, and we focus on geometric properties of several CAU structures. All CAUs have the same mass of 32 t. Hexacone has exceptionally high mass density near the leg tips, which helps to increase the moment of inertia. The moment of inertia of a Hexacone with respect to the horizontal pivot axis at the bottom line of the units is also the largest of the four tested. Hexacone is the most resistant to external torques when standing on its own. There is a possibility that a layer of Hexacones could be the most stable of the four types of units, especially when Hexacones are randomly placed or regularly placed with mixed vertical and horizontal columns. Future development of CAUs will aim to achieve a larger moment of inertia, raising the interlocking level and strengthening member endurance at the same time.