Cuboids, Rectangular Prisms and Cubes

A cuboid is a box-shaped object.

It has six flat sides and all angles are right angles.

And all of its faces are rectangles.

It is also a prism because it has the same cross-section along a length. In fact it is a rectangular prism.

Examples of Cuboids

Cuboids are very common in our world, from boxes to buildings we see them everywhere. We can even fit them inside other cuboids!


A box with a slot
cut as a handle

Cuboids in a
cuboid room

Boxes for model trains

Now that's just silly!

Volume and Surface Area

The volume of a cuboid is found using the formula:

Volume = Length × Width ×  Height

Which is usually shortened to:

V = l × w × h

Or more simply:

V = lwh

Surface Area

And the surface area is found using the formula:

A = 2wl + 2lh + 2hw

 

Example: Find the volume and surface area of this cuboid.

 
V = 10×5×4 = 200

A = 2×4×5 + 2×5×10 + 2×10×4
   = 40+100+80 = 220
 

Square Prism

When at least two of the lengths are equal it can also be called a square prism.

(Note: this doesn't stop us from calling it a rectangular prism if we want!)

Cube

When all three lengths are equal it is called a cube (or hexahedron)
and each face is a square.

A cube is still a prism.

And a cube is one of the Platonic Solids.

 
So:
  • A cube is just a special case of a square prism, and
  • A square prism is just a special case of a rectangular prism, and
  • They are all cuboids!

Note: The name "cuboid" comes from "cube" and -oid (which means "similar to, or resembling") and so says "it is like a cube".

Another use of -oid is when we talk about the Earth being a spheroid (not exactly a sphere, but close).