Spatial Reasoning Flashcards

If a cube is painted on all six faces and then cut into smaller cubes, how many of the smaller cubes will have paint on exactly two faces?

Answer: 12

Cubes with paint on exactly two faces are located along the edges of the original large cube, but not at the corners. A cube has 12 edges. If the large cube is cut into a 3x3x3 array of smaller cubes, each edge will have (3-2)=1 smaller cube with paint on exactly two faces. Therefore, 12 edges multiplied by 1 cube per edge equals 12 cubes.

Which shape will the net of a cube form when folded?

Answer: Cube

A net is a two-dimensional pattern that can be folded to form a three-dimensional shape. The net of a cube is specifically designed with six square faces arranged in a way that, when folded along its edges, it perfectly encloses a space to create a cube. This is the fundamental definition and purpose of a cube's net.

If a pyramid has a square base and is viewed from above, what shape will be visible?

Answer: Square

When a pyramid with a square base is viewed directly from above, the observer looks down onto its base. The apex (the top point) of the pyramid will appear as a central point, and the four triangular faces will converge towards it, but the outermost boundary of what is seen will be the square shape of its base.

Which three-dimensional shape will you get if you rotate a rectangle around one of its sides?

Answer: Cylinder

When a rectangle is rotated around one of its sides, that side becomes the axis of rotation. The opposite side sweeps out a circular path, forming the top and bottom circular bases of the resulting three-dimensional shape. The other two sides of the rectangle trace out the curved lateral surface, thus creating a cylinder.

If you have a solid block in the shape of a rectangular prism and you remove a smaller rectangular prism from one corner, how many faces will the resulting shape have?

Answer: 7

A rectangular prism initially has 6 faces. When a corner is removed, for example by making a single planar cut (a truncation), one new face is created. The three original faces that met at the corner are reshaped, and the three faces opposite the corner remain untouched. This results in a total of 3 untouched faces + 3 reshaped original faces + 1 new face = 7 faces for the resulting shape.

Which of the following is a characteristic of a regular tetrahedron?

Answer: Four triangular faces

A regular tetrahedron is a polyhedron composed of four equilateral triangular faces, with three meeting at each vertex. It is the simplest of all convex regular polyhedra and is also known as a 3-simplex. Its defining characteristic is having four identical triangular faces.

Which shape can completely fill a three-dimensional space without gaps when tiled together?

Answer: Cube

A cube is one of the few three-dimensional shapes that can tessellate, or completely fill, a three-dimensional space without leaving any gaps or overlaps when stacked together. This property is due to its perfectly flat faces and 90-degree angles, allowing for seamless packing in all directions. Other shapes like spheres or cylinders would leave significant gaps.