A cube is a three-dimensional figure that has six square faces, twelve edges, and eight vertices. It is referred to as a regular hexahedron and is a key concept in the study of solid geometry, which deals with studying shapes and figures in three dimensions, including surface areas and volume. In this article, we shall discuss the properties of a cube and how to find its surface area and volume, all of which is necessary for success in U.S. competitive exams such as the SAT, ACT, GRE, and GMAT.

Cube

A cube is a solid three-dimensional object which has six square faces, eight vertices, and twelve edges. It is one of the simplest shapes in three-dimensional space. All the six faces of a cube are squares, two-dimensional shapes. The length, breadth, and height are of the same measurement in a cube since the 3D figure is made up of squares that have all sides of the same length.

In the cube, the faces share a common boundary called the edge which is considered the bounding line of the face. The structure is defined with each face being connected to four edges, each vertex connected with three edges and three faces and each edge is in touch with two faces and two vertices.

Cube Shape

To understand the cube’s structure, let us first understand basic definitions like edge, vertex, and face, which are essential parts of these three-dimensional objects.

Face: The flat square surface of the three-dimensional object is known as the face of that object.

Vertex: A point where more than two lines meet is a vertex in a three-dimensional object. A vertex is also known as a corner.

Edge: An edge consists of a line segment joining two vertices.

A cube is the only regular hexahedron and is one of the five platonic solids having six faces, 12 edges, and eight vertices. A cube is also known as a square parallelepiped, an equilateral cuboid, and a right rhombohedron. Finally, a cube can be considered as a special case of the square prism.

Area of the cube

Now let’s understand how to calculate the cube’s total and lateral surface area.

The lateral area of a cube is the sum of areas of the faces of the cube excluding the top and bottom faces. There are four side faces, so the sum of areas of all four side faces of a cube is its lateral area. The lateral area of a cube is also known as its lateral surface area (LSA), and it is measured in square units.

The total surface area of the cube is the sum of the area of all the six faces of the cube. Since all the faces of the cube are made up of squares of the same dimensions, then the total surface area of the cube will be six times the surface area of any of the faces.

As we know, the cube has six square faces. Now, let us assume that the cube’s edge is x units. So x is the length of the edge of a square face in a cube.

Area of one face = Area of a square\( = x^{2}\)

The lateral surface area of cube\( = 4 \times\)area of one face

{LSA\(=4 x^{2}\(\)}

Now total surface area = LSA + area of the top and bottom faces

\( TSA= 6 x^{2} \)

Surface area of the cube \(=6 x^{2}\) squared units

Volume of the cube

Volume is the quantity of the space enclosed within a closed surface. There are different formulas based on various parameters to calculate the volume of the cube. It can be calculated using a measure of the cube’s side or diagonal and expressed in cubic units of length or length. That’s why there are two different formulas to find the volume of the cube;

The volume of a cube (based on diagonal)\(=\frac{\sqrt{3}\times d^{3}}{9}\)

where d is the length of the diagonal of the cube

The volume of a cube (based on side length)=\(s^{3}\)

where s is the length of the side of a cube

Diagonals of a cube

The diagonal length can be specified using the diagonal of a cube formula. A cube’s diagonal is a line segment that joins two opposite vertices of a cube. Each face diagonal forms the hypotenuse of the right-angled triangle. For example, a cube has 6 faces. On each face, 2 diagonals and 4 body diagonals connect the cube’s opposite vertices.

The length of the main diagonal of the cube \(=\sqrt{3}s\(\) units, where \(\)s\(\) is the length of each side of a cube

Length of face diagonal of cube\(\)=\sqrt{2}s\(\) units, where \(\)s\(\) is length of each side of a cube.

Cube Net

The cube net is formed when the square faces in a 3D figure are flattened by splitting at the edges, making it into a 2D figure. It means different orientations in which the faces of a cube can be opened and aligned on a flat surface.

Through the net of a cube, we can see the 6 square faces that combine at the edge to form a cube. There is a possibility of 11 different nets in a cube, as shown in the following figure.

For more information about the net of the cube, refer

Properties of cube

A cube is a square prism having important properties as follows:

1. A cube has 12 edges, six faces, and eight vertices
2. The angles between two faces or surfaces are \(\)90^{\circ}\(\)
3. The length, breadth, and height are the equal in a cube.
4. Each vertice in a cube meets 3 faces and 3 edges.
5. Each face in a cube meets four different faces.

Solved Examples of Cube

Q1.Determine the surface area of the cube having side \(\)2\frac{1}{2}\(\) inches

Answer: As here we know that side, \(\)x=2\frac{1}{2}\(\)

Surface area is given by , Surface area \(\)=6x^{2}\(\) \(\)

\begin{aligned}

&=6 \times\left(2\frac{1}{2}\right)^{2\\

&=6 \times(2.5)^{2}\\

&=37.5 \text{ squared inches }

\end{aligned}

\)

Q2.Find the volume of the cube whose side is \(8 \mathrm{~cm}\(\)

Answer: As here we have given that side,\(\)x=8\)

Volume is given by,Volume \(=x^{3}\)

\(=(8)^{3}\)

\(=512\) cubic cm

Q3.Find the length of an edge of a cube whose total surface area if 600 squared cm

Answer:As here we have given that area is 600 squared \(\mathrm{cm}\(\)

The total surface area of a cube \(\)=6x^{2}\)

\(

\begin{gathered}

\Rightarrow 6 x^{2}=600 \\

x^{2}=\frac{600}{6} \\

x^{2}=100 \\

x=10 \mathrm{~cm}

\end{gathered}

\)

Edge of the cube is \(10 \mathrm{~cm}\)

Q4. The side length of a cube is 5 m.What is the length of its diagonal across one of the faces?

Answer: As here we know all the sides of the cube are equal and it makes right angles with adjacent side.So here by using pythagoras theorem, we can calculate the length of the diagonal.

\(x^{2}+a^{2}=d^{2}\)

\(

\begin{gathered}

\Rightarrow a^{2}+a^{2}=d^{2}\\

\Rightarrow 5^{2}+5^{2}=d^{2}\\

\Rightarrow 25+25=d^{2}\\

\Rightarrow d^{2}=50\\

\Rightarrow d=\sqrt{50}\\

\Rightarrow d=\sqrt{25\times2}\\

\Rightarrow d=5\sqrt{2}

\end{gathered}

\)

Hence, \(5\sqrt{2}\) is the length of its acrossed faces diagonal.

To have a good hold on solid geometry concepts that find their way on U.S. competitive exams like the SAT, ACT, GRE, and GMAT, knowledge of the cube's properties, surface area, volume, and diagonals becomes necessary. When computing surface areas, finding volumes, or cracking problems related to diagonals, a good knowledge of these can really enhance the speed and precision of problem solving. As you get ready for these exams, work on applying these concepts to an array of questions to develop confidence and efficiency. Continue to explore and practice to enhance your mathematical foundation and score better on test day!