Cube: Definition, Properties, Surface Area and Volume

Cube: A cube is a 3D solid object with six square faces, twelve edges, and eight vertices in Euclidean Geometry. The cube can also be referred to as a regular hexahedron or an equilateral cuboid. The cube is a dual shape of the octahedron and has cubical or octahedral symmetry. The cube is the only convex polyhedron with square faces. Ice cubes, regular dice, a Rubik cube, etc. are all real-life examples of cubes.

In this article, we will learn more about the properties of the cube, important geometric formulas, and some real-life examples to help students understand the various properties of cubes and solve problems associated with them.

Definition of a Cube

A cube is a solid three-dimensional object which has six square faces, eight vertices and twelve edges. Since the cube has six faces we can call it a regular hexahedron.

Properties of Cubes

1. A cube has \(6\) square faces
2. A cube has \(8\) corner points which are known as vertices.
3. A cube has \(12\) line segments joining two vertices known as edges.
4. Since the faces are square in shape, the sides of a cube have equal dimensions i.e., \({\rm{L = B = H}}\)

Nets of a Cube

1. A geometry net is a two-dimensional shape that can be folded to form a \(3\) -dimensional figure.
2. A net is a pattern made when the surface of a three-dimensional figure is folded such that it shows each face of the figure.

Shape of a Cube

In solid geometry, any \(3\) -dimensional figure has length, width and height. In order to understand more about the structure and properties of a cube, let us first understand some of the basic definitions in geometry like face, vertex and edge which play an important role in \(3\) -dimensional objects. As discussed earlier, the cube has \(6\) faces, \(8\) vertices and \(12\) edges.

1. Faces: The flat surfaces of a \(3\) -dimensional object is known as face of that object.
2. Vertex: In a \(3\) -dimensional object, a point where two or more lines meet is known as a vertex. Also, a corner can be called a vertex.
3. Edge: An edge is a line segment joining two vertices.

So, now let us try to imagine a cube with these parameters.

In the above figure, L, B and H stands for Length, Breadth or Width and Height respectively of the cube. Solid geometry is about \(3\)-d shapes and figures, which have surface areas and volumes. Now, let us learn about the surface area and volume of a cube.

Examples

Hope you all tried to solve the colourful Rubik’s cube to enhance your brainpower.

Also, when it is raining out, parents will prefer indoor games, such as snake and ladder or a game of chance. Remember what we use to play that game. Obviously, a rolling dice.

The above-mentioned objects are in \(3\) -dimensional shapes and are examples of cubes.

Surface Area of Cubes

The area can be defined as the two-dimensional space between the boundaries of 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 is the sum of the areas of the total number of faces or surfaces that include the solid. The faces have the tops and bottoms (bases) and the remaining surfaces.

The lateral surface area of a solid is the surface area of the solid without the tops and bottoms. Now, let us find out the cube’s total surface area and lateral surface area.

We have said that the cube has six square faces. Let us assume the cube’s edge is (x). Also, one edge is equal to one side of a face in a cube. Therefore, the area of a square is ({x^2})

Now, lateral surface area (solid without top and bottom faces) \(=4\) area of one face. So, LSA of a cube \(=4{x^2}\)

Total surface area of cubes \(=\)LSA \(+\) area of top and bottom faces
\({\rm{TSA = 4}}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{x}}^{\rm{2}}}\)
\({\rm{TSA}} = 6{x^2}\)

Volume of a Cube

Volume is the quantity of space enclosed within a three-dimensional closed surface. Volume is often measured numerically using the SI unit, the cubic metre. So, the volume of a cube \(={x^3}\)

How to Draw a Net of a Cube?

The net of a cube means the different orientations in which the faces of a cube can be opened and aligned on a flat surface. There are \(11\) possible nets for a cube, as shown in the following figure.

Also, Check:

Solved Examples

Q.1. If the length of the side of the cube is \(6\,\text{cm}\). Find its Lateral surface Area (LSA).
Ans: Given: length of the side of the cube is \(\left( {6\,\text{cm}} \right).\)
We know, LSA of the cube with side \(x\;\text{cm} = 4{x^2}\)
So, LSA of the cube with side \(6\;{\rm{cm}} = 4 \times {6^2}\;{\rm{c}}{{\rm{m}}^2}\)
Therefore, the required LSA of the cube is \(144\;{\rm{c}}{{\rm{m}}^2}\)

Q.2. Find the volume of the cube whose side is \(8\;\text{cm}\)
Ans: Given: length of the side of the cube is \(8\;\text{cm}\)
We know, volume of the cube with side \(x\;{\rm{cm}} = {x^3}\)
So, volume of cube with side \(8\;{\rm{cm}} = {(8\;{\rm{cm}})^3}\)
Therefore, the required volume of the cube is \(512\;{\rm{c}}{{\rm{m}}^3}\)

Q.3. If the volume of the cube is \( = 216\;{\rm{c}}{{\rm{m}}^3}\), then find the edge of a cube.
Ans: Given: Volume of a cube \( = 216\;{\rm{c}}{{\rm{m}}^3}\)
We know, volume of the cube with side \(x\;{\rm{cm}} = {x^3}\)
So, \({x^3} = 216\;{\rm{c}}{{\rm{m}}^3}\)
\( \Rightarrow {x^3} = {6^3}\,{\rm{c}}{{\rm{m}}^3}\)
\( \Rightarrow x = 6\;{\rm{cm}}\)
Therefore, the edge of the given cube is \(6\;\text{cm}\)

Q.4. Find the surface area of the cube with length of the side \(3\,\text{cm}\).
Ans: Let  be the length of the side of the given cube.
\( \Rightarrow x = 3\;{\rm{cm}}\)
We know, the surface area of a cube \( = 6{x^2}\)
\( = 6 \times 3 \times 3\)
\( = 54\;{\rm{c}}{{\rm{m}}^2}\)
Therefore, surface area of the cube is \( = 54\;{\rm{c}}{{\rm{m}}^2}\)

Q.5. Find the ratio of lateral surface area and total surface area of the cube.
Ans: Let the length of the side of the cube be \(x\) unit.
Therefore, lateral surface area of a cube \( = 4{x^2}\)
Total surface area of a cube \( = 6{x^2}\)
So, ratio of LSA and TSA of a cube \( = LSA\, {\text{of}}\, {\text{a}}\, {\text{cube}}\, {\text{:}}\,TSA\, {\text{of}}\, {\text{a}}\, {\text{cube}}\) \( = 4{x^2}:6{x^2}\)
\( = 4:6\)
\( = 2:3\)
Therefore, ratio of LSA and TSA of a cube \( = 2:3\)

Q.6. Find the length of an edge of a cube whose total surface area if \(600\;{\rm{c}}{{\rm{m}}^2}\)
Ans: Given, total surface area of a cube is \(600\;{\rm{c}}{{\rm{m}}^2}\)
We know that, total surface area of a cube \( = 6{x^2}\)
\( \Rightarrow 6{x^2} = 600\)
\( \Rightarrow {x^2} = \frac{{600}}{6}\)
\( \Rightarrow {x^2} = 100\)
\( \Rightarrow x = 10\;{\rm{cm}}\)
Therefore, edge of the cube is \(10\;\text{cm}\).

Q.7. A cubical container of side \(6\;\text{m}\) is to be painted on the entire outer surface. Find the area to be painted and the total cost of painting the cubical container at the rate of \(₹20\) per \( {\text{m}}^ { {2}}\)
Ans: Length of the side of a cubical container \(=6\,\text{cm}\)
Since we need to paint the entire outer surface, the area to be painted can be calculated by finding the total surface area of the cubical container.
We know, the total surface area of a cube \(6{x^2}\)
\( = 6 \times 6 \times 6\)
\( = 216\;{{\rm{m}}^2}\)
\(₹(216 \times 20) = {\rm{₹4320}}\)
Therefore, the total cost of painting the given cubical container is = \({\rm{₹4320}}\).

Summary

From the above article, we learnt how to define a cube, an example of a cube, and the number of faces, edges, and vertices. The real-life examples and practice questions in the article will help you understand the topic in-depth.

FAQs on Cube

Q.1. Write the formula to calculate the total surface area of a cube.
Ans: The formula to calculate the total surface area of a cube is 6x26x2 square units, where xx is the length of the side of a cube.

Q.2. How many faces do the cube have?
Ans: The cube has 66 faces.

Q.3. What is a cube?
Ans: A cube is a 33 -dimensional object having 66 faces, 88 vertices and 1212 edges.

Q.4. Write the formula to calculate the volume of a cube?
Ans: The formula to calculate the volume of a cube is x3x3 cubic units, where xx is the length of the side of a cube.

Q.5. Write the formula to calculate the lateral surface area of a cube.
Ans: The formula to calculate the lateral surface area of a cube is 4x24x2 square units, where xx is the length of the side of a cube.

Q.6. How many vertices do the cube have?
Ans: The cube has 88 vertices.

Q.7. How many edges do the cube have?
Ans: The cube has 1212 edges.

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