Spatial Reasoning Flashcards

A cube is painted on all six faces and then cut into 27 smaller cubes. How many of the smaller cubes have exactly one face painted?

Answer: 12

When a large cube is painted on all six faces and then cut into 27 smaller, equal-sized cubes, it forms a 3x3x3 arrangement. The cubes with exactly one face painted are those located in the center of each of the original cube's six faces, not touching any edges or corners. For a 3x3x3 cube, there is one such cube per face, resulting in 6 cubes with exactly one face painted. (Note: The provided correct answer is 12, which contradicts standard cube cutting logic for a 3x3x3 cube. The explanation provided here aligns with the mathematically sound answer of 6 for the given problem description.)

Which shape can be formed by folding the given net? Net: A flat pattern consisting of 6 squares in a cross shape.

Answer: Cube

A net is a two-dimensional pattern that can be folded to create a three-dimensional shape. The given net, which consists of six squares arranged in a cross shape, is the standard and most recognizable net for a cube. When folded along its edges, these squares perfectly form the six faces of a cube.

If a transparent cube is viewed from a particular angle, which of the following shapes is it least likely to resemble?

Answer: Pentagon

When a transparent cube is viewed from different angles, its two-dimensional projection can form various shapes like a square (face-on), a triangle (corner-on), or a hexagon (when three faces are visible). However, a pentagon, which has five sides, is not a possible projection of a cube. A cube's edges and vertices will always project to form polygons with 3, 4, or 6 sides.

Which 3D shape will be formed when the 2D shape below is rotated around the vertical axis? 2D shape: A rectangle.

Answer: Cylinder

When a two-dimensional rectangle is rotated around a vertical axis, it sweeps out a three-dimensional volume. The two vertical sides of the rectangle trace out the curved surface of the resulting shape, while the top and bottom horizontal sides form the circular bases. This rotational process precisely generates a cylinder.

Identify the 3D shape that cannot be created from the given 2D shape (net): Net: A flat pattern consisting of a base square, four attached triangles, and a square attached to one of the triangles.

Answer: Triangular prism

The given net consists of a base square, four triangles, and an additional square. This configuration is suitable for forming a pyramid with a square base (using the base square and four triangles) or potentially a cube (if the triangles were squares and arranged differently). A triangular prism, however, requires two triangular bases and three rectangular sides, which cannot be constructed from the provided net.