# Cuboid

A cuboid is a space shape with six rectangular faces. Its angles are all right angles and its opposite faces are congruent and parallel to each other.

## What is a cuboid

A cuboid is a three-dimensional figure made up of six rectangular faces. The opposite faces (3 pairs) of a cuboid are congruent and all of the angles are right angles. A cuboid is also classified as a convex polyhedron. The name "cuboid" means "like a cube." Depending on the dimensions of the cuboid, it may be referred to as a cube or a variety of other names, as detailed below:

- Rectangular prism - a rectangular prism is another term for a cuboid, given that all angles in the rectangular prism are right angles.
- Hexahedron - a hexahedron is a polyhedron with 6 faces. A hexahedron is not necessarily a cuboid, but a cuboid is always a hexahedron.
- Cube - a cube is a cuboid in which all the edge lengths are equal. All faces of a cube are congruent.
- Right square prism - a right square prism is a cuboid in which at least two faces of the cube are squares, and all of the angles measure 90°. A right square prism is not necessarily a cube since the other 4 faces may be rectangles. Also, the reason we specify right square prism is because a square prism can have angles that are not necessarily 90°; In this case, the square prism would not be a cuboid.

Cuboids are very common in everyday life. A box, a refrigerator, and a book are all real life cuboid examples:

### Cuboid vs rectangular prism

A cuboid and a rectangular prism are very similar. In fact, a cuboid is always a rectangular prism, but a rectangular prism is not always a cuboid. Both have the same number of faces, edges, and vertices (6, 12, and 8 respectively), but the key difference is that a rectangular prism is only a cuboid if all the angles at the vertices of the cuboid measure 90°. If the angles at the vertices are not right angles, then a rectangular prism is not a cuboid. The figure below shows two rectangular prism examples: one that is a cuboid (left) and one that is not a cuboid (right).

As we can see from the figure, the only difference is that a rectangular prism only needs to have 2 bases that are rectangles, but a cuboid must have faces that are all rectangles.

## Characteristics of a cuboid

A cuboid is made up of faces, edges, and vertices. Its faces are further categorized as lateral faces (the 4 side faces) and bases (the top and bottom faces). Also, an edge of a cuboid is a line segment formed by the intersection of two adjacent faces while a vertex of a cuboid is the point of intersection of three edges.

### How many faces does a cuboid have?

A cuboid has 6 rectangular faces. The opposite pairs of faces in a cuboid are congruent and parallel to each other.

### How many edges does a cuboid have?

A cuboid has 12 edges. The opposite edges of a cuboid are congruent and parallel to each other. There are 3 groups of parallel edges in a cuboid, each of which consists of 4 edges. In a cuboid, any of the edges that intersect are perpendicular to each other.

### How many vertices does a cuboid have?

A cuboid has 8 vertices. The angle at each vertex of a cuboid measures 90°. Thus, adjacent faces of a cuboid are perpendicular. If this is not the case, the space figure may be classified as a rectangular prism, but not a cuboid.

### Cuboid properties

The following are some other cuboid properties.

- Face diagonals - a cuboid has 12 face diagonals (2 on each face).
- Space diagonals - a cuboid has 4 space diagonals connecting opposite pairs of vertices.
- All angles formed at the vertices of a cuboid measure 90°. As such, adjacent faces of a cuboid are perpendicular to each other.

## Cuboid formulas

Below are formulas for finding the total surface area, lateral surface area, volume, face diagonal length, space diagonal length, and perimeter of a cuboid. All of the formulas use the following figure as a reference.

In the figure above, length (l), width (w), and height (h) of the cuboid are shown. The height of a cuboid is the length of one of its edges that is perpendicular to the bases. The length and width are adjacent edges of one of the bases as shown above. Usually, the longer dimension of the base is considered the length while the shorter dimension is considered the width.

### Surface area of a cuboid

Below are the formulas for the total surface area and lateral surface area of a cuboid.

#### Total surface area of a cuboid

The total surface area, S_{T}, of a cuboid is

#### Lateral surface area of a cuboid

The lateral surface area of a cuboid is the total surface area of the cuboid minus the surface area of the bases. Referencing the above figure, the lateral surface area is the surface area of all the unshaded parts of the cuboid. The formula for the lateral surface area is therefore:

### Volume of a cuboid

The volume, V, of a cuboid is

where l is the length, w is the width, and h is the height of the cuboid.

### Length of face diagonals

The length of a face diagonal of a cuboid can be found using the Pythagorean theorem since all the vertices in a cuboid have angle measures of 90°. For example, the length of the space diagonals of the bases of a cubid can be found as:

Similarly, the length of the diagonals of the lateral faces can be found as,

or

L1, L2, and L3 are a reference to the three pairs of congruent faces in a cuboid: the 2 bases, and the 2 other pairs of lateral faces. Note that if the cuboid is a cube, we would only need to find the length of one of the face diagonals, since they would all be equal. The figure below shows the face diagonals of a cuboid in orange, red, and pink. Note that each of the face diagonals shown has a matching pair on the opposing face of the cuboid, so only half of the face diagonals of the cuboid are shown in the figure.

### Length of space diagonals

The space diagonal of a cuboid is a diagonal connecting two opposite vertices of the cuboid. A cuboid has 4 space diagonals. The formula for the length of the space diagonal of a cuboid is:

The figure below shows the space diagonals of a cuboid:

### Perimeter of a cuboid

The perimeter of a cuboid is the sum of all of the edges of a cuboid. A cuboid has 3 groups of 4 congruent edges (the length, width, and height). Thus, the perimeter of a cuboid formula is:

## Cuboid net

The net of a three-dimensional figure is the "unfolded" version of the figure. In other words, the net of a three-dimensional figure is the two-dimensional shape that can be modified such that it forms the three-dimensional figure. There can be multiple nets for the same three-dimensional figure. Below is one example of a cuboid net:

The lateral faces and the bases of the cuboid are labeled in the figure above. If we fold each lateral face up and close the top off with the second base, we transform the cuboid net into a cuboid.