Cuboid

Introduction

A cuboid is a solid shape or a three-dimensional shape in geometry. A cuboid is an 8 vertices, 12 edges, and 6 rectangular faces convex polyhedron. A cuboid is also known as a rectangular prism. An example of a cuboid in the real world is a rectangular box.

What is cuboid?

Imagine a shape that results from stacking numerous congruent rectangles on top of one another. The resultant shape is known as a cuboid.

Dimensions of a Cuboid

In order to describe the length, width (breadth), or height of an edge of a cuboid shape, it should be emphasised that there is no universal formula. The cuboid's horizontal face is assumed to have two dimensions: length and width, with length being the larger of the two. The height of a cuboid is thought to correspond to the length of any vertical edge when it is set flat on a table. The letters "l," "w," and "h" stand for the cuboid's length, width (breadth), and height, respectively. A cuboid's edge is the portion of a line joining two adjacent vertices, its vertex is the location where two or more edges meet, and its face is its flat surface.

Cuboid Faces Edges Vertices

There is a set number of faces, edges, and vertices on every 3D shape. A cuboid has six faces, twelve edges, and eight vertices. A cuboid has 2 top and bottom faces in addition to its 4 lateral faces. They are all rectangle-shaped. It features 12 edges, including 4 connecting edges and 8 edges on the top and bottom sides. The top and bottom faces' vertices make up its additional 8 vertices. At each vertex, three segments from each of the three dimensions come together.

Cuboid Formulas

Consider the following table to show the fundamental cuboid formulas. A cuboid has three main dimensions: l, w, and h..

Face Diagonals	$\mathrm{\sqrt{(l^{2}\:+\:w^{2})}\:units}$
Space Diagonals	$\mathrm{\sqrt{(l^{2}\:+\:w^{2}\:+\:h^{2})}\:units}$
Perimeter	$\mathrm{4(l\:+\:w\:+\:h)\:units}$

Diagonals of a Cuboid

A cuboid has two different kinds of diagonals because it is a three-dimensional shape −

• Diagonal faces

• Spatially Diagonal

Look at the following illustration, which depicts a cuboid's face and space diagonals.

Face Diagonal

The opposite vertices of a certain cuboid face can be connected to form face diagonals, but only two diagonals can be formed on a single cuboid face. A cuboid can be drawn with a total of 12 face diagonals since it has 6 faces.

Space diagonal

A line segment connecting a cuboid's opposing vertices is known as a space diagonal. The interior of the cuboid is traversed by the space diagonals. So it is possible to draw 4 space diagonals inside of it.

Surface Area of Cuboid

A cuboid's surface area is the whole area that the shape takes up. The surface area of a cuboid will depend on its length, breadth, and height because it is a three-dimensional object. Total surface area and lateral surface area are two different types of surface areas that it may have. As a result, the following formulas are provided to determine a cuboid's surface area:

Cuboid's overall surface area is equal to $\mathrm{S\:=\:2\:(lb\:+\:bh\:+\:lh)\:square\:units.}$

$$\mathrm{l\:=\:2h(l\:+\:b)\:square\:units\:is\:cuboid\:Lateral\:Surface\:Area}$$

where,

l = Length,

b = Breadth,

h = Height,

S = Total surface area, and

L = Lateral surface area

Volume of Cuboid

The volume of a cuboid is used to determine how much space it occupies. A cuboid's length, breadth, and height determine its volume. As a result, altering any one of these values alters the shape's volume. The cubic unit is used to express the volume of a cuboid. So, the following is the formula to get the cuboid's volume −

Hence, the $\mathrm{volume\:of\:a\:cuboid\:,\:V\:=\:l\times\:b\times\:h\:=\:lbh\:cubic\:units.}$

where,

l = Length

b = Breadth, and

h = Height

Cuboid Properties

The key characteristics of a cuboid make it simple to recognise one. These are what they are −

• A cuboid has 6 faces, 8 vertices, and 12 edges

• Only straight angles are created at the vertices of a cuboid

• The shapes of all the faces are rectangles.

• Two diagonal lines can be drawn on each face of a cuboid.

• The edges on either side of each other are parallel.

• A cuboid has the following measurements: length, breadth, and height.

A Few Examples of Cuboids from Real Life

We've compiled a list of some real-world cuboids. Look over the list to see if you can find any of these when you’re out and try to see if you have got your own to add

• Buildings − Take a look at the buildings as you stroll around a town or metropolis. You'll note that many of them have cuboidal shapes.

• Boxes − Pay attention to the boxes when you browse the supermarket, pick out a new pair of shoes, or unwrap a gift. The majority of the boxes are cuboid in shape, as you'll see.

Conclusion

In this tutorial, we learned about cuboids the properties of a cuboid. A cuboid is a rectangular prism. The Volume of a cuboid is given by $\mathrm{l\times\:b\times\:h}$, and the Lateral surface area is given by $\mathrm{2(l\:+\:b)\times\:h}$ and Total surface area is given by, $\mathrm{2(lb\:+\:bh\:+\:hl)}$.

FAQs

1. What is the diagonal measurement of a cuboid with dimensions of 7 by 6 by 3 units?

A cuboid formula has a diagonal length of $\mathrm{\sqrt{(l^{2}\:+\:w^{2}\:+\:h^{2})}}$ units.

Assume that length (l) = 7, width (w), = 6, and height (h), = 3

As a result, the diagonal's length is equal to $\mathrm{\sqrt{(7^{2}\:+\:6^{2}\:+\:3^{2})}}$ units.

Therefore, $\mathrm{The\:length\:of\:diagonal\:=\:\sqrt{(7^{2}\:+\:6^{2}\:+\:3^{2})}}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\sqrt{(49\:+\:36\:+\:9)}\:units\:=\:\sqrt{94}\:units}$

2. David owns a cuboidal box with measurements of 12 by 7 by 5. What is the box's volume?

Cuboid volume = length, width, and height in cubic units. By changing the variables provided, we obtain

$\mathrm{Volume\:=\:12\times\:7\times\:5\:cubic\:inches}$

$\mathrm{Volume\:=\:420\:cubic\:inches}$

3. Determine the cuboid's surface area. Its dimensions are 8 by 6 by 4 units?

The formula for a cuboid's surface area is 2(lw + wh + lh) square units. Let's calculate the length (l) at 8 units, the width (w) at 6 units, and the height (h) at 4 units using the provided dimensions.

Therefore, the surface area $\mathrm{=\:2\:\lbrace\:(8\times\:6)\:+\:(6\times\:4)\:+\:(8\times\:4)\rbrace}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:2(48\:+\:24\:+\:32)\:square\:units\:=\:2(104)\:=\:208\:square\:units.}$

4. What distinguishes a cuboid from a rectangular prism?

Since both will have 6 rectangular faces, 8 vertices, and 12 edges, the cuboid and rectangular prism are identical. Furthermore, both have a box-like appearance.