What is the maximum number of shapes that can be formed by cross sections of a cube? a.5 b.6 c.7 d.8

There are 7 different shapes can be made by cross sections of a cube, so option c is correct.

SOLUTION:

Given that, we have to find maximum number of shapes that  can be formed by cross section of a cube.

Now, let us see all the possibilities.

A square cross section can be created by slicing the cube by a plane parallel to one of its sides. An equilateral triangle cross section can be obtained by cutting the cube by a plane defined by the midpoints of the three edges emanating from any one vertex. One way to obtain a rectangle that is not a square is by cutting the cube with a plane perpendicular to one of its faces (but not perpendicular to the edges of that face), and parallel to the four, in this case, vertical edges. Pick a vertex, let's say A, and consider the three edges meeting at the vertex. Construct a plane that contains a point near a vertex (other than vertex A) on one of the three edges, a point in the middle of another one of the edges, and a third point that is neither in the middle nor coinciding with the first point. Slicing the cube with this plane creates a cross section that is a triangle, but not an equilateral triangle; it is a scalene triangle. Notice that if any two selected points are equidistant from the original vertex, the cross section would be an isosceles triangle. To get a pentagon, slice with a plane going through five of the six faces of the cube.   To get a hexagon, slice with a plane going through all six faces of the cube. To create a non-rectangular parallelogram, slice with a plane from the top face to the bottom. The slice cannot be parallel to any side of the top face, and the slice must not be vertical. This allows the cut to form no 90° angles. One example is to cut through the top face at a corner and a midpoint of a non-adjacent side, and cut to a different corner and midpoint in the bottom face.

Hence, there are 7 different shapes can be made, so option c is correct.

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