Prisms: parallel and perpendicular faces — Mathematics, 11–13
Find the parallel and perpendicular faces of a prism, and see why every cube is also a rectangular cuboid and a right prism.
Faces that never meet, faces that meet square
A prism has two equal bases, which are parallel, joined by side faces. Parallel faces never meet, however far you extend them. Perpendicular faces meet along an edge and make a right angle. In a cube or a rectangular cuboid there are 3 pairs of parallel faces, and every face is perpendicular to 4 others. In a right prism, each side face is perpendicular to both bases.
Each name adds one condition
Names of solids are nested, like boxes inside boxes. Right prism: the side faces are rectangles. Rectangular cuboid: a right prism whose bases are rectangles. Cube: a rectangular cuboid with all edges equal, so all 6 faces are squares. Each name keeps everything the names above it say, and adds one more condition. That is why a cube is also a cuboid and also a right prism.
Three solids, three names
A: six squares 4 cm by 4 cm. It is a cube, so also a cuboid and a right prism. B: two squares 4 by 4 and four rectangles 4 by 7. It is a cuboid, not a cube (4 ≠ 7). C: two triangles and three rectangles. It is a right prism, not a cuboid (triangular bases). Parallel pairs: A has 3, B has 3, C has 1 (the triangles). In A, 6 faces give 15 pairs; 15 − 3 = 12 are perpendicular.
A cube is not “only” a cube
Many people think each solid has one name only, as if “cube” ruled out “cuboid”. But a cube meets every condition of a rectangular cuboid, so it is one. The reverse is false: a box 4 by 4 by 7 is a cuboid but not a cube, because 4 ≠ 7. The nesting only reads downwards: every cube is a cuboid, every cuboid is a right prism, every right prism is a prism, and never the other way round.
Boxes, rooms and packaging
A shoebox is a cuboid: lid and bottom are parallel faces, and the walls stand perpendicular to the floor. In a classroom, floor and ceiling are parallel, and each wall is perpendicular to the floor. A tent shaped like a triangular prism has two parallel triangles at the ends, and the rectangular sides are perpendicular to them. A die is a cube: opposite faces are parallel.
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