Shadow Blocks

Assuming the two cuts are made so that their normals are on the same plane. (That is, both cuts are "left-to-right", instead of one "left-to-right" and one "front-to-back".)

Without loss of generality, suppose the cylinder is $x^2 + y^2 \le 1, 0 \le z \le 3$ , and the cuts have normal on the $xz$ -plane. The shadows are the cross sections of each piece with the $xz$ -plane. Each shadow is in the form of a trapezoid whose bases are on the line $x = -1, y = 0$ and on the line $x = 1, y = 0$ . The height of the trapezoid is constant, 2 (because it's the distance between the two lines). Since the shadows have equal areas, this means the sum of the bases are constant. The sum of the bases over all shadows is 6 (each line contributes 3 units length, $0 \le z \le 3$ ), so each shadow's sum of bases is 2. This means, the intersection of each shadow with the center $x = 0, y = 0$ has length 1. Mark the points $(0, 0, 1), (0, 0, 2)$ ; each cut goes through one of these points (the lower cut passing through $(0,0,1)$ and the upper cut passing through $(0,0,2)$ ).

Now, take the plane $z = 1$ . The lower piece has some portion above this plane; rotate it 180 degrees around the axis $y = 1, z = 1$ . It can be easily shown that this makes a perfect cylinder of height 1, so the volume of the piece is simply the volume of this cylinder, $\pi$ . Similar with the upper piece, cutting the plane $z = 2$ and rotating through the axis $y = 1, z = 2$ . The middle piece has the remaining volume, $\pi$ , so all three pieces have the same volume.

My solution is a lot shorter tbh, so I'm not entirely sure it's correct. Anyway, there you go: So the volume of a cylinder is calculated, by taking the fundamental form and multiplying it by the height. So if all the shadows have the same area, if reshaped into a rectangle (with the broad of the circumference of the cylinder of course), they also have the same height. I hope you get what I mean.

All trapezoids have the same height: the distance of two opposite, parallel line segments on the surface of the cylinder (2r). The areas of the three trapezoids are equal. The area of a trapezoid is the midsegment multiplied by its height, so the midsegments of the three trapezoids must be equal.

The volume of a cylindrical segment (when the cut planes don't cut the bases as defined above) is $\frac{h1+h2}{2}$ $r^2\pi$ , where h1 and h2 is the minimum and maximum heights of the cylindrical segment. Note that the midsegment of a shadow trapezoid is $\frac{h1+h2}{2}$ . So the volume of one cylindrical section is $mr^2\pi$ , which is the same for all bodies as their m and r values are equal.