Rectangular Prism

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What is a Rectangular Prism?

In geometry, a rectangular prism is a solid object which has identical bases with flat faces, and same cross-section throughout its length. A rectangular prism is a solid box-shaped 3-dimensional object whose faces are all rectangles. A rectangular prism has a total 6 flat faces and all angles are at right angles as shown in the figure below.

A rectangular prism consists of 2 rectangular bases and 4 rectangular sides. The bases are congruent and parallel to each other. Since every prism is a polyhedron (solid with flat faces), hence it does not consist of any curved sides. The same is applied to a rectangular prism and hence it is also a polyhedron with no curved sides. A rectangular prism has 8 vertices, 6 faces and 12 edges as shown in the figure below.

We can find plenty of examples of rectangular prisms in the world, including the very room we are sitting in. Some of the examples are skyscraper buildings, a room, a box etc.

Diagonal of a Rectangular Prism:

The diagonal of a rectangular prism is the line drawn to connect the opposite vertices. Finding the diagonal length is very useful in various calculations. We can calculate the diagonal of a rectangular prism if we know the measurements of the prism’s length, width and the height.

Let the length be = l, width = w, and height = h.

Then the diagonal of a rectangular prism, d = √(l2 + w2 + h2).

Example: How much is the diagonal length of a rectangular prism whose length is 7m, width is 4m and height is 3m?

Given that length l = 7m, width w = 4m and height, h = 3m.
Diagonal of a rectangular prism, d = √(l2 + w2 + h2).
Hence applying the above formula, we get: d = √(72 + 42 + 32).
This gives us d = √(49 + 16 + 9) = √74.
Now, √74 can be approximated to its decimal value as 8.6m.
Hence 8.6m is the diagonal length of the given rectangular prism!

Lateral Area of a Rectangular Prism:

Lateral Area of a rectangular prism is the area of the faces of the prism, excluding its bases. This implies that we actually consider the area of only 4 faces of the prism and do not calculate the area of the 2 bases of the prism. This only gives us the lateral area of the rectangular prism.

We can find the lateral area of the rectangular prism using the formula as shown below:

Lateral Area of a Rectangular Prism = Perimeter of the Base * Height
L = P * h

Since we are finding lateral area of a rectangular prism, hence the perimeter of the base is nothing but the perimeter of the base rectangle. We know that the perimeter of a rectangle is the sum of all its sides. This gives us that the Perimeter of a rectangle, P = 2l + 2w (where l = length and w = width) of the rectangular prism.

Hence the Lateral Area of a Rectangular prism can now also be written as:

Lateral Area of a Rectangular Prism = Perimeter of the Base * Height

L = P * h
L = (2l + 2w) * h
Or L = 2lh + 2wh
(l = length, w = width, h = height)

Example: Calculate the lateral area of a rectangular prism if given that the length is 7m, width is 4m, and height is 3m.

Given that length l = 7m, width w = 4m and height, h = 3m.
Lateral area of the rectangular prism, L = Perimeter of the base * Height
L = 2lh + 2wh ==> L = (2* 7* 3) + (2* 4 * 3) ==> L = 42 + 24 = 66
Hence, the Lateral Area, L = 22m * 3m = 66m2 or 66 square meters.

Surface Area of a Rectangular Prism:

Surface area of a rectangular prism is the surface area of all the faces of the prism. Since surface area includes the area of all the 6 faces of the prism, hence this is also known as the Total Surface Area of a rectangular prism.

In order to calculate the surface area of a rectangular prism, we use the formula as shown below:

Surface Area of a Rectangular Prism = Lateral Area + (2 * Area of the Base)
S = L + (2 * B)

Since the Lateral area is 2lh + 2wh, we can plug-in its formula in the place of ‘L’. Now, the area of the base is nothing but the area of the base rectangle. Area of a rectangle = length * width. Therefore, the area of the base, B = l * w

Hence the Surface Area of a Rectangular prism can now also be written as:

Surface Area of a Rectangular Prism = Lateral Area + (2 * Area of the Base)
S = L + (2 * B)
(Since L = 2lh + 2wh and B = l*w)
S = 2lh + 2wh + 2lw
(l = length, w = width, h = height)

Example: Calculate the surface area of a rectangular prism if given that the length is 6m, width is 3m and height is 5m.
 
Given that length l = 6m, width w = 3m and height, h = 5m.
Surface Area of the rectangular prism, S = 2lh + 2wh + 2lw
Hence, S = (2* 6* 5) + (2* 3* 5) + (2 * 6 * 3) ==> S = 60 + 30 + 36 = 126m2

Volume of a rectangular prism:

Volume of a rectangular prism can be defined as the space available inside the prism. To understand this, we can take an example of a room. The space available within the room tells us about its volume. Similarly the space available within a rectangular prism gives us its volume.
 
In order to find the Volume of a rectangular prism, we can use the below formula:

Given the length = l, width = w and height = h
Volume of a rectangular prism, V = length * width * height
V = l * w * h

Example: Calculate the volume of a rectangular prism if given that the length is 6m, width is 3m and height is 5m.

Given that length l = 6m, width w = 3m and height, h = 5m.
Volume of the rectangular prism, V = l * w * h
Hence, Volume, V = 6m * 3m * 5m = 90m2
 
A rectangular prism, also known as a cuboid is a prism consisting of rectangles as its surfaces. It is a 3-dimensional geometric shape just like any other prism, since it consists of length, width and height. A rectangular prism has its bases as rectangles and hence its name. The total surface area of a rectangular prism is the sum of the areas of all the surfaces and the volume of a rectangular prism is the measure of space contained inside the prism.

Example 1: Calculate the volume of the rectangular prism whose length is 4m, width is 1.5m and height is 3m.

Given the length of the rectangular prim, l = 4m

Width of the rectangular prism, w = 1.5m

Height of the rectangular prism, h = 3m

Volume of a rectangular prism, V = (length) * (width) * (height)

Hence, volume of the rectangular prism, V = 4m * 1.5m * 3m = 18m3

Therefore, the volume of the given rectangular prism = 18m3

Example 2: Calculate the surface area of a rectangular prism given its length as 4m, width as 1.5m and height as 3m.

Given the length of the rectangular prim, l= 4m

Width of the rectangular prism, w= 1.5m

Height of the rectangular prism, h= 3m

Total Surface area of the rectangular prism, A = 2lw + 2lh + 2wh

Hence Area, A = (2* 4* 1.5) + (2* 4* 3) + (2* 1.5* 3)

Area, A= 12+ 24+ 9= 45

Hence, Surface area of the rectangular prism, A = 45m2

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