About Rectangular Prisms
What are Rectangular Prisms?
In geometry, a rectangular prism is a three dimensional solid figure. It has a rectangular base and top and four parallelogram shaped faces.
A notebook, rectangular aquarium and a writing board are some of the real life examples of objects that are shaped like a rectangular prism, besides the examples mentioned earlier like a gift box or a box of tissues. A rectangular prism is often called a cuboid as well.
Properties of a Rectangular Prism
- A rectangular prism has 6 faces, 12 edges and 8 vertices.
- The top and base of a rectangular prism are rectangles.
- Opposite sides or edges of a rectangular prism are congruent and parallel to one another.
What are the types of Rectangular Prisms ?
Rectangular prisms are of two types:
- Right rectangular prism
- Oblique rectangular prism
When the lateral faces of a prism are perpendicular to the rectangular base and top surface, it is called a right rectangular prism. In other words, if the lateral surfaces are rectangular in shape, the prism is a right rectangular prism.
If the lateral faces and rectangular base and top are not perpendicular, this type of prism is called an oblique rectangular prism. In other words, the lateral faces of an oblique prism are in the shape of parallelograms.
What is the net of a Rectangular Prism ?
In order to find the surface area of any solid object, we can use a two dimensional representation of the solid that is known as its net. The net of a rectangular prism is the two-dimensional representation of the prism in which the faces of the prism are laid out flat.
What is the Surface Area of a Rectangular Prism?
A rectangular prism has two types of surface area formulas: total surface area (TSA) and lateral surface area (LSA).
The total surface area is the sum of the areas of all the faces of a rectangular prism. The area of the faces can be calculated by using the net of the prism.
Total surface area = sum of areas of the faces
= lw + lw + wh + wh + hl + hl
(where, l = length, w = width, and h = height)
= 2 (lw + wh + hl)
The lateral surface area is the sum of the areas of all the side faces (excluding the base and top) of the rectangular prism. The formula is given by:
Lateral surface area = sum of areas of side faces
= wh + wh + hl + hl
(where, l = length, w = width, and h = height)
= 2 (wh + hl)
The surface area is expressed in square units.
What is the Volume of a Rectangular Prism?
The volume of a solid rectangular prism represents the space occupied within the prism. The measurement of volume is expressed in cubic units. The formula is given by:
Volume = area of rectangular base \( \times \) height of prism
= base area (b) \( \times \) height (h)
= l \( \times \) w \( \times \) h
Note : The volume of oblique prisms is calculated in the same way as the volume of the right prism.
Solved Rectangular Prism Examples
Example 1: Find the volume of a rectangular prism whose length, width, and height are 7 cm, 4 cm, and 5 cm respectively.
Solution:
Given, length (l) = 7 cm
Width (w) = 4 cm
Height (h) = 5 cm
The formula for the volume of a rectangular prism is,
V = Length \( \times \) Width \( \times \) Height
= 7 \( \times \) 4 \( \times \) 5
= 140 cm\( ^3 \)
Example 2: Albert bought a pair of shoes and the shoe box is in the shape of a rectangular prism. The length of the shoe box is 5 in, width is 3 in, and the height is 4 in. Find the total surface area of the shoe box.
Solution:
The dimensions of the shoe box is given as, l = 5 in, w = 3 in, h = 4 in.
The box is in the shape of a rectangular prism so according to the formula for surface area of rectangular prism,
Total surface area = 2 (lw + wh + hl)
TSA = 2 (5 \( \times \) 3 + 3 \( \times \) 4 + 4 \( \times \) 5) [Substitute the values of l, w and h]
= 2 (15 + 12 + 20)
= 2 \( \times \) 47
= 94 sq. inches
Example 3: Susan is packing a gift in a rectangular box whose dimensions are 18 in, 12 in, and 7 in and it will be wrapped up by a gift wrapper. How much gift wrapper is needed to wrap the gift box by Susan?
Solution:
The dimensions of the gift box are given as length (l) = 18 in, width (w) = 12 in, and height (h) = 7 in.
In order to find the amount of gift wrapper, we have to find the total surface area of the gift box.
Total surface area = 2 (lw + wh + hl)
Substitute the values of l,w and h and solve,
= 2 (18 \( \times \) 12 + 12 \( \times \) 7 + 7 \( \times \) 18)
= 2 (216 + 84 + 126)
= 2 (426)
= 852 in\( ^2 \)
Therefore the amount of gift wrapper required for wrapping the gift box is 852 in\( ^2 \).
Examples of rectangular prisms that we find in daily life are notebooks, calculators, bricks, laptops, and so on.
A rectangular prism can also be known as a cuboid.
Even though both cubes and cuboids are prisms, a cube has six faces with each of these faces being a square, while a cuboid has six faces that are rectangular in shape.
A right rectangular prism is termed as a cuboid. It has six rectangular faces, 12 edges and eight vertices. The lateral faces of a right rectangular prism are perpendicular to the base and top of the prism.