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July 28, 202239 Insightful Publications

**Cuboid Definition: **Cuboid shaped objects surround us in our day-to-day life. From television sets, books, carton boxes to bricks, mattresses, and shoeboxes, cuboid objects are all around us. In Geometry, a cuboid is a three-dimensional figure with six rectangular faces, twelve edges, and eight vertices. The cuboid shape comes with a closed three-dimensional figure surrounded by rectangular faces, which are plane regions of rectangles.

This article discusses cuboids, including their definitions and components, like their surface area and volume. Also, we will learn about the net of a cuboid. Read on to find out more.

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**Definition**: A cuboid is a solid three-dimensional object which has six faces that are rectangular in shape with eight vertices, and twelve edges. Since the cuboid has six faces we can call it a regular hexahedron.

The textbooks we read, the lunch boxes we bring to school, the mattresses we sleep on and the bricks we use to build a house, and many other things, are well-known examples of cuboids in our environment.

As discussed earlier, the cuboid has \({\rm{6}}\) faces, \({\rm{8}}\) vertices, and \({\rm{12}}\) edges.

In solid geometry, any \({\rm{3}}\)-dimensional figure has length, breadth, and height.

Let’s first learn about the basic parameters like face, vertex, and edge which play an important role in \({\rm{3}}\)-dimensional objects.

**Faces:** Any of the individual flat surfaces of a solid object is known as the face of that object.

**Vertex:** In a \({\rm{3}}\)-dimensional object, a point where two or more lines meet is known as a vertex. Also, a corner can also be referred to as a vertex.

**Edge:** An edge is a line segment joining two vertices.

So, now let us try to imagine a cuboid with the above-said parameters. The shape of the cuboid appears to be as follows:

In the figure below, \({\rm{l,}}\,{\rm{b}}\) and \({\rm{h}}\) stand for length, breadth or width, and height respectively of the cuboid.

Solid geometry is associated with \({\rm{3 – D}}\) shapes and figures with surface areas and volumes.

Now, let us learn about the surface area and volume of a cuboid.

Let us look at some of the formulas for cuboid:

We know that, area can be defined as the space enclosed by a flat shape or the surface of an object. The area of a figure is the number of unit squares that enfold the surface of a closed figure.

Similarly, the total surface area of a solid is the sum of the areas of the total number of faces or surfaces of the solid.

The lateral surface area of a solid is the surface area of the solid excluding the top and base.

Now, let us find out the total surface area and a lateral surface area of a cuboid.

We have said that the cuboid has six rectangular faces.

In the above figure, let \({\rm{l}}\) be the length, \({\rm{b}}\) be the breadth, and \({\rm{h}}\) be the height of a cuboid.

Therefore, \(AD = BC = GF = HE = l\)

\(AB = CD = GH = FE = b\)

\(CF = DE = BG = AH = h\)

Now, the lateral surface area of the cuboid \(= \) Area of rectangular face \(ABGH + \) Area of rectangular face \(DCEF + \) Area of rectangular face \(ADEH + \) Area of rectangular face \(BCGF\)

\( = \left( {AB \times BG} \right) + \left( {DC \times CF} \right) + \left( {AD \times DE} \right) + \left( {BC \times CF} \right)\)

\( = \left( {b \times h} \right) + \left( {b \times h} \right) + \left( {l \times h} \right) + \left( {l \times h} \right)\)

\( = 2\left( {b \times h} \right) + 2\left( {l \times h} \right)\)

\( = 2\,h\left( {l + b} \right)\)

Therefore, lateral surface area of a cuboid \( = 2h\left( {l + b} \right)\,{\rm{sq}}{\rm{.units}}\)

Now, the total surface area of a cuboid is the sum of the areas of a total number of faces or surfaces that include the cuboid. The faces include the top and bottom (bases) and the remaining surfaces.

Area of face \(ABCD = \) Area of face \(EFGH = \left( {l \times b} \right)\)

Area of face \(CDEF = \) Area of face \(ABGH = \left( {b \times h} \right)\)

Area of face \(BGCF = \) Area of face \(ADHE = \left( {l \times h} \right)\)

\( = \)Area of \(\left( {ABCD + EFGH + CDEF + ABGH + BGCF + ADHE} \right)\)

\( = \left( {l \times b} \right) + \left( {l \times b} \right) + \left( {b \times h} \right) + \left( {b \times h} \right) + \left( {l \times h} \right) + \left( {l \times h} \right)\)

\( = 2\left( {l \times b} \right) + 2\left( {b \times h} \right) + 2\left( {l \times h} \right)\)

\( = 2\left( {lb \times bh \times lh} \right)\)

Volume is the amount of \(3\)-dimensional space occupied by a solid object, such as the space occupied or contained by a substance (solid, liquid, gas, or plasma). Volume is often measured numerically using the SI-derived unit, the cubic meter \({{\text{m}}^3}\).

So, volume \({\rm{ = }}\left( {{\rm{Length \times Breadth \times Height}}} \right)\) \( = l \times b \times h\)

Therefore, the volume of a cuboid \( = lbh\)

Perimeter is the sum of lengths of all the edges of a cuboid.

From the above figure, we know that,

\(AD = BC = GF = HE = l\)

\(AB = CD = GH = FE = b\)

\(CF = DE = BG = AH = h\)

So, perimeter of a cuboid \( = AD + BC + GF + HE + AB + CD + GH + FE + CF + DE + BG + AH\)

\( = \left( {l + l + l + l} \right) + \left( {b + b + b + b} \right) + \left( {h + h + h + h} \right)\)

\( = 4\left( {l + b + h} \right)\)

Therefore, the perimeter of a cuboid \( = 4\left( {l + b + h} \right)\)

The formula to find the length of the diagonal of a cuboid is given by \(\sqrt {{l^2} + {b^2} + {h^2}} .\)

1. A geometry net is a two-dimensional shape that can be folded to form a \(3\)-dimensional shape or a solid.

2. A net is a pattern made when the surface of a three-dimensional figure is put forward flat showing each face of the figure.

3. A solid may have different nets.

A cuboid can be drawn in \(54\) different nets of three different lengths.

One of them is shown below:

Let us look at some of the properties of a cuboid:

1. A cuboid has \(6\) rectangular faces.

2. A cuboid has \(8\) corner points which are known as vertices.

3. A cuboid has \(12\) line segments joining two vertices known as edges.

4. All angles in a cuboid are right angles

5. The edges opposite to each other are parallel.

Let us look at some of the solved examples for cuboid:

*Question 1: Find the total surface area (TSA) of a cuboid whose length, breadth, and height are \({\rm{6}}\,{\rm{cm,}}\,{\rm{4}}\,{\rm{cm}}\) and \({\rm{2}}\,{\rm{cm}}\) respectively.***Answer:** Given: \(l = 6\,{\rm{cm}},\,b = 4\,{\rm{cm}}\) and \(h = 2\,{\rm{cm}}\)

We know that the total surface area of a cuboid is \(2\left( {lb + bh + lh} \right)\)

So, the total surface area of the given cuboid \( = 2\left[ {\left( {6 \times 4} \right) + \left( {4 \times 2} \right) + \left( {2 \times 6} \right)} \right]{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

\({\rm{ = 2}}\left[ {{\rm{24 + 8 + 12}}} \right]{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

\({\rm{ = 2}}\left( {{\rm{44}}} \right)\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

\({\rm{ = 88}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

Therefore, the total surface area of cuboid \({\rm{ = 88}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

*Question 2: Find the total surface area of the following cuboid whose length, breadth and height are \({\rm{4}}\,{\rm{cm,}}\,{\rm{4}}\,{\rm{cm}}\) and \({\rm{10}}\,{\rm{cm}}\) respectively.***Answer:** Given: \(l = 4\,{\rm{cm}},\,b = 4\,{\rm{cm}}\) and \(h = 10\,{\rm{cm}}\)

We know that the total surface area of a cuboid is \(2\left( {lb + bh + lh} \right).\)

So, the total surface area of the given cuboid \( = 2\left[ {\left( {4 \times 4} \right) + \left( {4 \times 10} \right) + \left( {4 \times 10} \right)} \right]{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

\( = 2\left[ {16 + 40 + 40} \right]{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

\( = 2\left( {96} \right){\rm{c}}{{\rm{m}}^{\rm{2}}}\)

\({\rm{ = 192}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

Therefore, the total surface area of cuboid \({\rm{ = 192}}\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

*Question 3: Calculate the lateral surface area of a cuboid of dimensions \({\rm{10}}\,{\rm{cm \times 6}}\,{\rm{cm \times 5}}\,{\rm{cm}}{\rm{.}}\)***Answer:** Given: \(l = 10\,{\rm{cm}},\,b = 6\,{\rm{cm}}\) and \(h = 5\,{\rm{cm}}\)

We know that the lateral surface area of a cuboid is \(2h\left( {l + b} \right)\) \({\text{sq}}{\text{.units}}\)

So, LSA of the given cuboid \( = 2 \times 5\left( {10 + 6} \right)\)

\( = 10\left( {16} \right)\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

\( = 160\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

Therefore, LSA of the given Cuboid \( = 160\,{\rm{c}}{{\rm{m}}^{\rm{2}}}\)

*Question 4: Find the length of the longest pole that can be put in a room of dimensions \({\rm{10}}\,{\rm{m \times 10}}\,{\rm{m \times 5}}\,{\rm{m}}\)***Answer:** Given: \(l = 10\,{\rm{m}},\,b = 10\,{\rm{m}}\) and \(h = 5\,{\rm{m}}\)

Length of the longest pole \( = \) Diagonal of a Cuboid (room)

We know that the diagonal of a Room \( = \sqrt {{l^2} + {b^2} + {h^2}} \)

\( = \sqrt {{{10}^2} + {{10}^2} + {5^2}} {\rm{m}}\)

\( = \sqrt {100 + 100 + 25} \,{\rm{m}}\)

\( = 225\,{\rm{m}}\)

\( = 15\,{\rm{m}}\)

Therefore, the length of the longest pole is \( = 15\,{\rm{m}}{\rm{.}}\)

*Question 5: A cuboid has dimensions \(60\;{\rm{cm}} \times 54\,{\rm{cm}} \times 30\,{\rm{cm}}.\) Find the Volume of a cuboid.***Answer:** Given: \(l = 60\,{\rm{cm}},\,b = 54\,{\rm{cm}}\) and \(h = 30\,{\rm{cm}}\)

We know that the volume of a cuboid \( = l \times b \times h\)

\( = \left( {60 \times 54 \times 30} \right){\rm{c}}{{\rm{m}}^{\rm{3}}}\)

\( = 97200\,{\rm{c}}{{\rm{m}}^{\rm{3}}}\)

Therefore, the volume of a cuboid \( = 97200\,{\rm{c}}{{\rm{m}}^{\rm{3}}}\)

From the above article, we learn how to define a cuboid, an example of a cuboid and what is the number of faces, edges and vertices of a cuboid. Also, we hope that we have helped you to learn how to find the volume and surface area of the cuboid with the given measures.

Also Check,

**Q.1. Do cuboids have square faces?**** Ans:** Yes, a cuboid can have a square face. The cuboid is a solid \(3\)-dimensional object having \(12\) rectangular faces. If any two faces of a cuboid are square, then it is called a square cuboid.

Example: Square Prism.

**Q.2. Can a cuboid have all rectangular faces?**** Ans:** Yes, the cuboid is a solid \(3\)-dimensional object having \(12\) rectangular faces.

**Q.3. What is the cuboid formula?**** Ans:** The formula to find the various parameters of a cuboid is given below.

The total surface area of a cuboid is \(2\left( {lb + bh + lh} \right)\,{\rm{sq}}{\rm{.units}}\)

The Lateral surface area of a cuboid is \(2\left( {lb + bh + lh} \right)\,{\rm{sq}}{\rm{.units}}\)

The length of a diagonal of a cuboid \( = \sqrt {{l^2} + {b^2} + {h^2}} \)

The volume of a cuboid \( = l \times b \times h\,{\rm{cubic}}\,{\rm{units}}\)

The perimeter of a cuboid \( = 4\left( {l + b + h} \right)\)