Geometry Problem Solver Rectangular cube
There is no such thing as a rectangular cube: a cube is already the shape whose faces are all squares. What people mean by the phrase is a rectangular prism, also called a cuboid, which is the box shape with six rectangular faces. A cube is the special case of it where all edges are equal.
Why the phrase exists
People reach for it because English has no everyday word for the box shape. Cube is the only 3D word most people know, so it gets stretched to cover anything box-like, and rectangular is bolted on to signal that it is not a perfect cube.
The formal names are rectangular prism, used in most school curricula, and cuboid, used in the UK and in much of the world. Both mean the same solid: six faces, all rectangles, meeting at right angles.
Volume and surface area
Volume is length times width times height. A box 8 by 5 by 3 holds 8 x 5 x 3 = 120 cubic units, and the order of multiplication does not matter.
Surface area is the total of the six faces. Opposite faces are identical, so it comes to 2(lw + lh + wh). For the same 8 by 5 by 3 box that is 2(40 + 24 + 15) = 158 square units.
For a cube both formulas collapse: volume is s cubed and surface area is 6s squared, because every face is the same square.
Cube against rectangular prism
| Cube | Rectangular prism | |
|---|---|---|
| Faces | 6 squares | 6 rectangles |
| Edges | 12, all equal | 12, in three groups of 4 |
| Volume | s³ | l x w x h |
| Surface area | 6s² | 2(lw + lh + wh) |
| Everyday example | A die | A shoebox |
Questions
Is a rectangular cube the same as a cuboid?
Yes, if by rectangular cube you mean the box shape. Cuboid and rectangular prism are the two correct names for it; rectangular cube is informal and self-contradictory.
Are all cubes rectangular prisms?
Yes. A square is a kind of rectangle, so a solid with six square faces satisfies the definition of a rectangular prism. The reverse does not hold.
What is the volume of a rectangular prism?
Length times width times height. Keep all three in the same unit and the answer comes out in that unit cubed.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
