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Rectangular Prism, also called a cuboid, is a polyhedron and its geometry is like that of a cuboid which is why it is also called a cuboid. A rectangular prism comprises six faces, and all the faces are in a rectangle shape and have twelve edges. Due to its cross-section along the length, it is said to be a prism. In mathematics, geometry deals with the concept of different dimensions of figures. A rectangular prism is a three-dimensional figure with six faces, twelve edges, and eight vertices, and that also has a surface area and volume. Here, we will be discussing the rectangular prism in detail, its formulae, properties along with some important questions.
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Key Takeaways: Rectangular prism definition, Rectangular prism concepts, Rectangular prism formulae, Rectangular prism types, Right rectangular prism
What is a Rectangular Prism?
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A rectangular prism is a three-dimensional figure with six faces(two faces are opposite to each other at the top and bottom and four faces are lateral to each other), eight vertices, and twelve edges. The faces of the prism are all rectangular and are such that the opposite faces are always congruent. Thus, it has three pairs of identical faces. A rectangular prism has a volume and a surface area. The rectangular prism is also called a cuboid.
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Types of Rectangular Prism
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The rectangular prism can be of two different types. They are as follows-
- Right Rectangular Prism
- Oblique Rectangular Prism
Right Rectangular Prism
The right rectangular prism is a prism with six faces that are all rectangles and all the angles are right angles.
Right Rectangular prism
Oblique Rectangular Prism
An oblique prism is a prism that has bases that are not perpendicular to each other. A rectangular prism that has faces that are not exactly aligned directly to each other.
Oblique Rectangular Prism
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Properties of Rectangular Prism
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A rectangular prism has some definite properties that are as follows-
- A rectangular prism has six faces, eight vertices, and twelve edges.
- It is a three-dimensional figure that has a length, width, and height.
- The faces are rectangular in a right rectangular prism. The faces are parallelograms in an oblique rectangular prism.
- The opposite faces in a rectangular prism are always congruent.
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Rectangular Prism Formulae
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A rectangular prism is a three-dimensional figure which has a surface area and volume. The dimensions of a rectangular prism length(l), width(w), height(h).
Volume of Rectangular Prism
The volume of a rectangular prism is the amount of space inside it. The volume of the rectangular prism is obtained by multiplying its base area by its height. Now,
- The base area of the rectangular prism \(= l \times w \)
- The height of the rectangular prism \(= h\)
Therefore, the volume of the rectangular prism, \(V = l \times w \times h\)
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Surface Area of Rectangular Prism
The surface area of a rectangular prism is of two types-
-
The total surface area of a rectangular prism is the sum of the areas of all the faces in the rectangular prism.
The TSA of the rectangular prism is given by the formula,
\(=2 (lw + wh + hl)\)
- The lateral surface area of a rectangular prism is the sum of the areas of all its side faces without the base faces.
The lateral surface area of the rectangular prism is given by the formula,
\(=2 (wh + hl)\)
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Nets of a Rectangular Prism
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The surface area of a rectangular prism can be calculated by calculating the area of the nets of a prism. The surface area of a prism is calculated from the nets.
Nets of the rectangular prism
When the prism is turned in a plane, all the sides of the prism are then visible. The surface area of the rectangle is calculated for each rectangle and then all the areas are added together to find the total surface area of the rectangular prism.
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Things to Remember
- A rectangular prism is a three-dimensional figure with six faces, eight vertices, and twelve edges.
- The faces of the prism are all rectangular and are such that the opposite faces are always congruent.
- For any given polyhedron like rectangular prism,
\(F + V - E = 2\)
where F stands for the number of faces, V stands for the number of vertices and E stands for the number of edges. This relationship is called Euler’s formula. - A rectangular prism is a three-dimensional figure which has a surface area and volume. The dimensions of a rectangular prism length(l), width(w), height(h).
The volume of the rectangular prism, \(V = l \times w \times h\) - A rectangular prism is a three-dimensional figure which has a surface area and volume. The dimensions of a rectangular prism length(l), width(w), height(h).
The TSA of the rectangular prism is given by the formula,
\(=2 (lw + wh + hl)\)
The lateral surface area of the rectangular prism is given by the formula,
\(=2 (wh + hl)\) - The surface area of a prism is calculated from the nets.
Sample Questions
Ques. Find the volume of a rectangular prism whose length, width, and height are 8cm, 6cm, and 4cm, respectively. [2 marks]
Ans. The data given are,
length, \(l = 8\ cm;\) width, \(w = 6\ cm;\) height, \(h = 4\ cm\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = l \times w \times h\) cubic units
\(V = 8 \times 6 \times 4\) cubic units.
\(V = 192\) cubic units.
Therefore, the volume of a rectangular prism is \(192\ cm^3\).
Ques. Ms. Jeniffer gives prizes to the students of her class for performing acts of kindness. She displays prizes like colorful pencils, fancy erasers, and color pens in a glass case, shaped more of a rectangular prism of length = 16 in, width = 5 in and height = 3 in. Find the total surface area of the prize case. [2 marks]
Ans. The prize case dimensions given are,
Length, \(l = 16\ in\)
Width, \(w = 5\ in\)
Height, \(h = 3\ in\)
So, the total surface area of the prize case,
TSA \(= 2 (lw + wh + hl)\)
\(= 2 (16 \times 5 + 5 \times 3 + 16 \times 3)\)
\(= 2 (80 + 15 + 48)\)
\(= 2 \times 143\)
\(= 286\ in^2\)
Ques. Find the surface area and volume of a rectangular prism with the length, width, and height are 8 cm, 6 cm, and 4 cm respectively. [2 marks]
Ans. The data given is,
length, \(l = 8\ cm;\)
width, \(w = 6\ cm;\)
height, \(h = 4\ cm\)
The formula used to find the total surface area of a rectangular prism is,
\(A = 2 (lw + wh + hl)\)
\(A = 2 (8 \times 4 + 6 \times 4 + 8 \times 6)\)
\(A = 2 (32 + 24 + 48)\)
\(A = 2 \times 104\ cm^2\)
\(A = 208\ cm^2\)
Ques. Find the area of a rectangular prism whose length, width, and height are given, respectively. [5 marks]
-
5 cm, 8 cm, 10 cm
-
6.2 cm, 4.4 cm, 9 cm
Ans. a) The data given is,
Length, \(l = 5\ cm\)
Width, \(w = 8\ cm\)
Height, \(h = 10\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (5 \times 10 + 8 \times 10 + 5 \times 8)\)
\(= 2 (50 + 80 + 40)\)
\(= 2 \times 170\ cm^2\)
\(= 340\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\) cubic units
\(= 5 \times 8 \times 10\) cubic units.
\(V = 400\ cm^3\)
b) The data given is,
Length, \(l = 6.2\ cm\)
Width, \(w = 4.4\ cm\)
Height, \(h = 9\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (6.2 \times 4.4 + 4.4 \times 9 + 9 \times 6.2)\)
\(= 2 (27.28 + 39.6 + 55.8 )\)
\(= 2 \times 122.68\ cm^2\)
\(= 245.36\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 6.2 \times 4.4 \times 9\) cubic units.
\(V = 245.52\ cm^3\).
Ques. Find the volume and total surface area of rectangular prism with the following dimensions. [4 marks]
-
3 cm x 4 cm x 5 cm
-
2.5 cm x 6 cm x 9 cm
Ans. a) The data given is,
Length, \(l = 3\ cm\)
Width, \(w = 4\ cm\)
Height, \(h = 5\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (3 \times 5 + 4 \times 5 + 3 \times 4)\)
\(= 2 (15 + 20 +12)\)
\(= 2 \times 47\ cm^2\)
\(= 94\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 3 \times 4 \times 5\) cubic units.
\(V = 60\ cm^3\)
b) The data given is,
Length, \(l = 2.5\ cm\)
Width, \(w = 6\ cm\)
Height, \( h = 9\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (2.5 \times 9 + 6 \times 9 + 2.5 \times 6)\)
\(= 2 (22.5 + 54 + 15 )\)
\(= 2 \times 91.5\ cm^2\)
\(= 183\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 2.5 \times 6 \times 9\) cubic units.
\(V = 135\ cm^3\)
Ques. Rahul has a chocolate box whose shape resembles a rectangular prism. Its dimensions are as follows, length 6 in, height 2 in, and width 4 in. Find the volume of the box. [3 marks]
Ans. The dimensions of the chocolate box given are,
length, \(l = 6\ in\)
width, \(w = 4\ in\)
height, \(h = 2\ in\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 6 \times 4 \times 2\) cubic inches.
\(= 48\ in^3\)
Ques. 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? [3 marks]
Ans. The dimensions of the given rectangular gift box are,
length, \(l = 15\ in\)
width, \(w = 10\ in\)
height, \(h = 8\ in\)
The rectangular box (rectangular prism) needs to be wrapped with gift paper therefore, the total surface area needs to be calculated because the wrapper will be spread across the total surface area of the prism to cover it completely. Thus,
The formula for Total Surface Area of a rectangular prism \(= 2 (lw + wh + hl)\)
\(= 2 (15 \times 10 + 10 \times 8 + 15 \times 8)\)
\(= 2 (150 + 80 + 120)\)
\(= 2 \times 350\)
\(= 700\ in^2\)
Ques. Olivia's mother surprised her with pasta packed in a rectangular prism-shaped lunch box. How much quantity of pasta is served to Olivia given that the dimensions of the rectangular prism are as follows, length = 5 in, width = 4 in, height = 3 in. [3 marks]
Ans. The dimensions of the rectangular prism-shaped lunch box are
length, \(l = 5\ in\)
width, \(w = 4\ in\)
height, \(h = 3\ in\)
Now, the amount of pasta that can be contained in the lunch box can be calculated by finding out the volume of the container. Thus,
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 5 \times 4 \times 3\) cubic inches.
\(= 60\ in^3\)
Ques. Jenny's grandmother handed over her beautiful hand-carved chest to him for storing his stuff, especially woolens. The chest is 3 ft long and 2 ft wide having a shape of that of a rectangular prism has a capacity of 12 cubic ft. Find its height. [2 marks]
Ans. The dimensions of the chest given are,
Length, \(l = 3\ ft.\)
Width, \(w = 2\ ft.\)
Height, \(h =\) to be found out
Volume, \(v = 12\) cubic ft.
Now the formula for calculating the volume can be used to calculate the height,
Volume, \(V = length \times width \times height\) cubic units
\(12 = l \times w \times h\)
\(12 = 3 \times 2 \times h\)
\(12 = 6 \times h\)
\(h = 2\ ft.\)
Ques. Find the surface area volume of the rectangular prism with the following dimensions. [4 marks]
-
14 cm x 8 cm x 5 cm
-
3.4 cm x 9.4 cm x 9.7 cm
Ans. a) The data given is,
Length, \(l = 14\ cm\)
Width, \(w = 8\ cm\)
Height, \(h = 5\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (14 \times 5 + 8 \times 5 + 14 \times 8)\)
\(= 2 (70 + 40 + 112)\)
\(= 2 \times 222\ cm^2\)
\(= 444\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 14 \times 8 \times 5\) cubic units.
\(V = 560\ cm^3\)
b) The data given is,
Length, \(l = 3.4\ cm\)
Width, \(w = 9.4\ cm\)
Height, \(h = 9.7\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (3.4 \times 9.7 + 9.4 \times 9.7 + 3.4 \times 9.4)\)
\(= 2 (32.98 + 91.18 + 31.96 )\)
\(= 2 \times 156.12\ cm^2\)
\(= 312.24\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 3.4 \times 9.4 \times 9.7\) cubic units.
\(V = 310.012\ cm^3\)
Ques. Find the area of a rectangular prism whose length, width, and height are given, respectively. [4 marks]
-
13 cm, 15 cm, and 6 cm
-
4.5 cm, 7.2 cm, 2.3 cm
Ans. a) The data given is,
Length, \(l = 13\ cm\)
Width, \(w = 15\ cm\)
Height, \(h = 6\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (13 \times 6 + 15 \times 6 + 13 \times 15)\)
\(= 2 (78 + 90 + 195)\)
\(= 2 \times 363\ cm^2\)
\(= 726\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 13 \times 15 \times 6\) cubic units.
\(V = 1170\ cm^3\)
b) The data given is,
Length, \(l = 4.5\ cm\)
Width, \(w = 7.2\ cm\)
Height, \(h = 2.3\ cm\)
The formula used to find the total surface area of a rectangular prism is,
Surface area, \(A = 2 (lw + wh + hl)\)
\(= 2 (4.5 \times 2.3 + 7.2 \times 2.3 + 4.5 \times 7.2)\)
\(= 2 (10.35 + 16.56 + 32.4)\)
\(= 2 \times 59.31\ cm^2\)
\(= 118.62\ cm^2\)
The formula that is used to find the volume of a rectangular prism is
Volume, \(V = length \times width \times height\) cubic units
\(= l \times w \times h\)
\(= 4.5 \times 7.2 \times 2.3\) cubic units.
\(V = 74.52\ cm^3\)
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