What is a rectangular prism
A rectangular prism is a three-dimensional (3D) figure that is made up of at least 2 rectangular faces and 4 parallelogram faces, or 6 rectangular faces. A prism is a 3D figure with 2 parallel bases of the same shape. The shape of the base determines the name of the prism. Thus, to be a rectangular prism, the bases must both be rectangular. The other faces of a prism, referred to as the lateral faces, can be rectangles, parallelograms, or squares.
Real life examples of rectangular prisms
A rectangular prism is a shape that is commonly seen in everyday life. A tissue box, an eraser, and an aquarium are all real-life rectangular prism examples:
Characteristics of a rectangular prism
Below are some rectangular prism characteristics and properties.
- A rectangular prism is made up of bases, edges, lateral faces, and vertices.
- Faces - a rectangular prism has 6 faces (2 bases and 4 lateral faces).
- Edges - a rectangular prism has 12 edges.
- Vertices - a rectangular prism has 8 vertices.
- Right rectangular prism - all the faces of a right rectangular prism are rectangles.
- Oblique rectangular prism - the lateral faces of an oblique rectangular prism are parallelograms; the bases are rectangles.
- Opposing pairs of faces in a rectangular prism are parallel and congruent. Opposing pairs of edges are also congruent and parallel.
- Any cross section that is parallel to a face of a rectangular prism has the shape of a rectangle or parallelogram and is congruent to the face. The figure below shows two cross sections, shaded in purple, that are parallel and congruent to the bases of the rectangular prism.
Types of rectangular prisms
There are two rectangular prism types: right rectangular prisms and oblique rectangular prisms.
Right rectangular prism
A right rectangular prism is a rectangular prism in which the bases are perpendicular to its lateral faces. In other words, all the faces meet at right angles (90°). As a result, all of the faces of a right rectangular prism are rectangles (this includes square bases or faces). The figure below shows a right rectangular prism example:
Oblique rectangular prism
An oblique rectangular prism is a rectangular prism in which the bases and lateral faces are not perpendicular. This results in the lateral faces being parallelograms rather than rectangles, while the bases are still both rectangles. The figure below shows an oblique rectangular prism example:
A cuboid is a 3D figure made up of 6 rectangular faces. All the faces of a cuboid meet at 90° angles. This is the key difference between a cuboid and a rectangular prism. In a rectangular prism, the lateral faces may be parallelograms which means that some of the faces do not meet at 90° angles. In other words, all cuboids are rectangular prisms, but not all rectangular prisms are cuboids. The figure below shows a cuboid example:
A cube is a special case of a cuboid that shares all of the properties of a cuboid while also having edges that are all congruent.
Rectangular prism formulas
Below are formulas for the volume, surface area, and space diagonals of a rectangular prism.
Volume of a rectangular prism
The rectangular prism volume formula is
where l is the length of the base, w is the width of the base, and h is the height of the prism. For a right rectangular prism, the height is its vertical edge. For an oblique rectangular prism, the height is the perpendicular distance from any point on one base to the other base.
Below is a rectangular prism volume example.
Find the volume of the rectangular prism shown in the figure below.
The length of the rectangular prism is 6, the width is 4, and the height is 9, so its volume is:
Note that volume is in cubic units. For example, if the edges are measured in feet, the volume of the rectangular prism is 216 ft3.
Rectangular prism surface area
The surface area of a rectangular prism can be broken up into two parts: lateral surface area and total surface area. The total surface area is the surface area of all of the faces of a rectangular prism and is given by the formula
where l is length, w is width, and h is height. Use the figure in the volume section above as reference.
The lateral surface area of a rectangular prism formula is:
Below is a rectangular prism surface area example showing how to find both lateral surface area and total surface area.
Find the surface area of the rectangular prism shown in the figure below.
The length of the rectangular prism is 6, its width is 4, and its height is 9. Recall that opposing faces of a rectangular prism are congruent. In the figure, one of each pair of congruent faces is shaded (the front face and the right face). The area of the front face is found by multiplying the length and the height to get 6 × 9 = 54. The area of the right face is found by multiplying width and height to get 4 × 9 = 36. Since these areas only represent half of the area, we multiply by 2 to take into account their congruent counterparts and find that the lateral surface area of the rectangular prism is:
To find the total surface area, we add the area of the two congruent bases. Each base has an area of 6 × 4 = 24, so the two have an area of 48. The total surface area is the sum of the areas of all the pairs of faces:
Surface area is measured in squared units, so given that each edge is measured in feet, the rectangular prism has a total surface area of 276 ft2.
Space diagonal of a rectangular prism
The rectangular prism space diagonal formula is
where l is length, w is width, and h is height. Refer to the figure in the volume section above. Below is an example of how to use the space diagonal of a rectangular prism formula.
Find the length of the space diagonals (the dotted purple lines) of the rectangular prism shown in the figure below.
The length of the rectangular prism is 6, the width is 4, and the height is 9, so the space diagonal of the rectangular prism is:
A rectangular prism has 4 congruent space diagonals, so each space diagonal in the above rectangular prism measures 11.533 ft, given that the edges are measured in feet.
Rectangular prism net
The net of a 3D figure is the "unfolded" version of the figure. In other words, the net of a 3D figure is the 2D shape that can be modified such that it forms the 3D figure. A given 3D figure can have multiple nets. Below is one example of a rectangular-prism net.