Lindsay's shape

After studying various 3D shapes and finding formulas for their volumes, I challenged my students to invent a new shape. Lindsay created a shape \((\)with height \(2 \text{ cm})\) that is circular at the top \((\)with radius \(1\text{ cm})\) but square at the bottom \((\)with side length \(2\text{ cm}).\) Lindsay created this shape from a paraboloid: sliced four times parallel to the paraboloid's axis and two times perpendicular to the paraboloid's axis. Lindsay's shape is pictured below.

Find the volume of Lindsay's shape in \(\text{cm}^{3},\) which can be written as \(\frac{A}{B}+C\pi\) with \(A,B,C\) integers, \(A\) and \(B\) coprime, and \(B\) positive.

Give the value of \(A+B+C.\)