A rectangular prism is a three-dimensional shape, having six faces, where all the faces (top, bottom, and lateral faces) of the prism are rectangles such that every two opposite side faces are identical. Like all three-dimensional shapes, a rectangular prism also has volume and surface area. A rectangular prism is also known as a cuboid. Let us learn more about a rectangular prism along with the formulas to find its volume and surface area.
|1.||What Is Rectangular Prism?|
|2.||Types of Rectangular Prisms|
|3.||Properties of Rectangular Prism|
|4.||Formulas of Rectangular Prism|
|5.||FAQs on Rectangular Prism|
What Is Rectangular Prism?
A rectangular prism is a prism whose bases (both its top face and bottom face) are rectangles. It has 6 faces in total out of which there are 3 pairs of identical opposite faces. i.e., every two opposite faces are identical in a rectangular prism. It has three dimensions, length, width, and height. Some examples of a rectangular prism in real life are rectangular tissue boxes, school notebooks, laptops, fish tanks, large structures such as cargo containers, rooms, storage sheds, etc. In the figure below, you can see how a rectangular prism looks like and how its net (the shape formed by it when its faces are opened) looks like.
Types of Rectangular Prisms
There are two types of rectangular prisms that are classified depending on the shape of the side faces (or) the angle made by the side faces with the base.
- Right rectangular prism
- Oblique rectangular prism
A right rectangular prism has its side faces perpendicular to each of its bases. In this, all side faces are rectangles. An oblique rectangular prism is a prism that is NOT a right rectangular prism and its side faces are parallelograms. In general, a rectangular prism without any specifications is a right rectangular prism.
Properties of Rectangular Prism
Here are the properties of a rectangular prism that follow its definition.
- A rectangular prism has 6 faces, 8 vertices, and 12 edges.
- Its base and top are always rectangles.
- The side faces are rectangles for a right rectangular prism whereas the side faces of an oblique rectangular prism are parallelograms.
- It has 3 dimensions, length, width, and height.
- Every two opposite faces of a rectangular prism are congruent.
Formulas of Rectangular Prism
In this section, we will learn the formulas of the volume and surface area of a rectangular prism. For both of these, let us consider a rectangular prism of length 'l', width 'w', and height 'h'. Along these dimensions, let us assume that 'l' and 'w' are the dimensions of the base. Here are the formulas for the volume and surface area of a rectangular prism.
Let us see how to derive these formulas.
Volume of Rectangular Prism
The volume of a rectangular prism is the space that is inside it. We know that the volume of any prism is obtained by multiplying its base area by its height. Here,
- The base area of the rectangular prism = lw (using the area of a rectangle formula)
- The height of the rectangular prism = h
Thus, the volume of the rectangular prism, V = lw × h = lwh.
Surface Area of Rectangular Prism
There are two types of surface area of a rectangular prism, one is the total surface area (TSA) and the other is the lateral surface area (LSA).
- TSA of a rectangular prism is the sum of the areas of all of its faces.
- LSA of a rectangular prism is the sum of the areas of all its side faces (no bases).
We can calculate the areas of the side faces of a rectangular prism using its net as shown below.
Using the above figure,
The total surface area (TSA) of a rectangular prism
= The sum of areas of all faces
= lw + lw + wh + wh + hl + hl
= 2 (lw + wh + hl)
The lateral surface area (LSA) of a rectangular prism
= The sum of areas of side faces
= wh + wh + hl + hl
= 2 (wh + hl)
We will see the applications of these volume and surface area formulas of a rectangular prism in the section below.
Solved Examples on Rectangular Prism
Example 1: Joe has a chocolate box whose shape resembled that of a rectangular prism whose length is 6 in, height is 2 in and width is 4 in. Find the volume of the box.
The dimensions of the given chocolate box are,
length, l = 6 in
width, w = 4 in
height, h = 2 in
Its volume is, V = lwh
= 6 × 4 × 2
= 48 in3
Answer: The volume of the box = 48 in3.
- Example 2: A gift is packed in a rectangular box (rectangular prism) of dimensions 15 in, 10 in, and 8 in and it needs to be wrapped with gift paper. How much gift paper is required to wrap the gift box?
The dimensions of the given gift box are,
length, l = 15 in
width, w = 10 in
height, h = 8 in
To find the amount of gift paper required, we need to find the total surface area of the box.
TSA of a rectangular prism = 2 (lw + wh + hl)
= 2 (15 × 10 + 10 × 8 + 8 × 15)
= 2 (150 + 80 + 120)
= 700 in2.
Answer: The amount (area) of the gift paper required = 700 in2.
FAQs on Rectangular Prism
What Is a Rectangular Prism?
A rectangular prism is a 3-d object which has 6 rectangular faces among which every two opposite faces are congruent. It has 8 vertices, 6 faces, and 12 edges.
What Is a Rectangular Prism Also Known As?
A rectangular prism resembles a cube, but it is not a cube. All its properties are the same as that of a cube except that its faces are rectangles (whereas the faces of a cube are squares). Thus, it has a name that is similar to a cube, which is a cuboid. So another name of a rectangular prism is a cuboid.
What Is the Difference Between a Cube and a Rectangular Prism (Cuboid)?
Both cube and cuboid are prisms. A cube has 6 faces all of which are identical squares whereas a cuboid (or a rectangular prism) has 6 faces all of which are rectangles among which opposite faces are identical.
What Is the Ratio of Corners to Faces in Rectangular Prisms?
A rectangular prism has 8 corners and 6 faces. So the ratio of the corners to faces in a rectangular prism is 8 : 6 (or) 4 : 3.
Why Is a Rectangular Prism Called a Polyhedron?
A polyhedron is a 3-d shape with flat faces. i.e., a polyhedron has all its bases to be polygons. A rectangular prism is a polyhedron as it has flat faces (or polygons).
What Is the Volume of a Rectangular Prism?
The volume of a rectangular prism is its base area multiplied by its height. Thus, the volume (V) of a rectangular prism of length 'l', width 'w', and height 'h' is V = lwh as its base area (area of the rectangle of length 'l' and width 'w') is lw.
What Is the Total Surface Area of a Rectangular Prism?
The total surface area of a rectangular prism is obtained by adding the areas of all its faces. Thus, the total surface area (TSA) of a rectangular prism of length 'l', width 'w', and height 'h' is TSA = 2 (lw + wh + hl).
What Is the Lateral Surface Area of a Rectangular Prism?
The lateral surface area of a rectangular prism is the sum of the areas of all its side faces. Thus, the lateral surface area (LSA) of a rectangular prism of length 'l', width 'w', and height 'h' is LSA = 2 (wh + hl).