A rectangular prism is a three-dimensional solid bounded by faces, edges, and vertices.
Faces: Has 6 rectangular faces, where 2 faces are the bases (top face and bottom face), and the other 4 are lateral faces. Each opposite face is congruent and parallel to the other.
Edges: Has 12 edges, which are the vertices of the top and bottom faces.
Vertices: Has 8 vertices, including 8 edges on the top and bottom faces and 4 edges that connect them.
Rectangular Prism
Due to its shape, a rectangular prism is also called a cuboid. Some real-life examples of rectangular prisms are a shoe box, an ice cream bar, or a matchbox.
A rectangular prism is of 2 types based on the angle formed by the lateral faces at the bases. They are right and oblique rectangular prisms.
Right and Oblique Rectangular Prisms
Right and Oblique Rectangular Prisms
Right Rectangular Prism – It has all the faces perpendicular. Thus, every face is rectangular. A cuboid is a right rectangular prism, as all its angles are right angles. Thus, a cuboid also holds all the above features except the oblique rectangular prism. The faces of a right rectangular prism are rectangles.
Oblique Rectangular Prism – Its lateral faces are not perpendicular to its bases. So, every lateral face is parallelogram-shaped. The faces of an oblique rectangular prism are parallelograms.
Formulas
Surface Area
The surface area of a rectangular prism is the entire space occupied by its outermost layer (or faces). It is expressed in square units such as m2, cm2, mm2, and in2. We can calculate 2 types of surface areas in a rectangular prism.
Lateral Surface Area
The lateral surface area (LSA) of a rectangular prism is only the sum of the surface area of all its faces except the bases. The formula to calculate the lateral surface area of a rectangular prism is given below:
Lateral Surface Area (LSA) = 2(wh + hl), here, l = length, w = width, h = height
Total Surface Area
The total surface area (TSA) of a rectangular prism is the sum of the lateral surface area and twice the base area. The formula to calculate the TSA of a rectangular prism is given below:
Total Surface Area (TSA) = 2(lw + wh + hl), here l = length, w = width, h = height
Let us solve an example to understand the concept better.
Find the lateraland totalsurface area of a rectangular prism with length 16 cm, width 12.5 cm, and height 10 cm.
Solution:
As we know, Lateral Surface Area (LSA) = 2(wh + hl), here l = 16 cm, w = 12.5 cm, h = 10 cm ∴ LSA = 2(12.5 × 10 + 10 × 16) = 570 cm2 Total Surface Area (TSA) = 2(lw + wh + hl), here l = 16 cm, w = 12.5 cm, h = 10 cm ∴ TSA = 2(16 × 12.5 + 12.5 × 10 + 10 × 16) = 970 cm2
Volume
The volume of a rectangular prism is its space in the three-dimensional plane. The formula to calculate the volume of a rectangular prism is given below:
Volume (V) = l × w × h, here l = length, w = width, h = height
Find the volume of a right rectangular prism. Dimensions of a rectangular prism are 9 cm long, 6 cm wide, and 15 cm high.
Solution:
As we know, Volume (V) = l × w × h, here l = 9 cm, w = 6 cm, h = 15 cm ∴ V = 9 × 6 × 15 = 810 cm2
Diagonals
There are two types of diagonals found in a rectangular prism:
Face Diagonals
A face diagonal is a line segment that connects two opposite corners (or vertices) on any face of a rectangular prism.
Each face of a rectangular prism has 2 diagonals. Since the prism has 6 faces, the total number of face diagonals is 6 × 2 = 12.
Space Diagonals
A space diagonal is a line segment that connects two opposite corners (or vertices) of a rectangular prism, passing through its interior. Therefore, each rectangular prism has 4 space diagonals.
Thus, each rectangular prism has 4 space diagonals.
The formulas to calculate the length of face diagonals and space diagonals of a rectangular prism are given below:
Face Diagonal = √(l2 + w2) units
Space Diagonal = √(l2 + w2 + h2) units, here l = length, w = width, h = height
Find the diagonal of a rectangular prism with dimensions of 6 cm, 5 cm, and 4 cm.
Solution:
As we know, The equation of the space diagonal of a rectangular prism is the same as that of a cuboid. ∴ Diagonal (D) = ${\sqrt{l^{2}+w^{2}+h^{2}}}$, here l = 6 cm, w = 4 cm, h = 5 cm ${\therefore D=\sqrt{6^{2}+4^{2}+5^{2}}}$ = 77 cm
Perimeter
The perimeter of a rectangular prism is the sum of the lengths of all the edges. The formula to calculate the perimeter of a rectangular prism is:
Perimeter of a rectangular prism = 4(l + w + h) units, here l = length, w = width, h = height
Calculate the perimeter of a rectangular prism with the dimensions 11 cm × 5 cm × 4 cm.
Solution:
As we know, The perimeter of a rectangular prism = 4(l + w + h) units, here l = 11 cm, w = 5 cm, h = 4 cm ∴ P = 4(11 + 5 + 4) = 80 cm
Nets
The net of a rectangular prism is the 2D shape we get when the 3D-shaped rectangular prism is unfolded. This view helps us to identify the sides of the rectangular prism.
A rectangular prism with three different lengths has 54 distinct nets. Here, a few different nets of a rectangular prism are shown.
