Surface Area of a Cube: Definition, Formula, Examples

Surface Area of a Cube: The surface area is defined as the sum of the areas of all closed surfaces or the total area of all faces of a three-dimensional object is known as the surface area of that three-dimensional object. In geometry, a cube is known as a three-dimensional figure with six square faces, twelve edges, and eight vertices.

Let us assume a three-dimensional object – a cube that needs to be painted on its outer surface. If we need to calculate the total cost required to paint that cube, then we need to find the sum of the areas of six surfaces of the cube. The total area of the six surfaces is called the total surface area of that three-dimensional object.

Surface Area of a Cube Definition

A cube is a solid three-dimensional object which has all of its six faces as squares.

The cube has \(6\) faces \(8\),and \(12\) edges.

Let us learn about the basic parameters like face, vertex, and edge, which play an important role in\(3 – \)dimensional objects.

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

Vertex: In a \(3 – \)dimensional object, a point where two or more lines meet is known as a vertex. Also, we can say a corner can be called a vertex.

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

Properties of a Cube

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 squares, the sides of a cube have equal dimensions, i.e., \(L\, = B = H\)

Examples of a Cube

We can see a lot of different cubical objects around us. They are Rubik’s cube, dice, sugar cubes, ice cubes, etc.

What Is the Surface Area?

The amount of outer space wrapping a three-dimensional shape is known as the surface area. The area of any three-dimensional geometric shape can be categorized into three types. They are:

1. Curved surface area
2. Lateral surface area
3. Total surface area

Curved Surface Area: The csa of a cube is the area of all the curved regions of the solid.

Lateral Surface Area: The lateral surface area refers to the surface area of a three-dimensional object that has polygonal faces excluding the top and bottom. It is the sum of the areas of all sides of a three-dimensional object except its top and bottom bases.

Total Surface Area: The tsa of a cube is the sum of all the faces including top, and bottom faces of the solid three-dimensional object.

Surface Area of a Cube

The total surface area of a cube is the sum of all \(6\) square faces of a cube. The faces include the top, bottom (bases) and the remaining surfaces.

The lateral surface area of a cube is the surface area of the cube without the top and bottom.

Total Surface Area of a Cube

The total surface area of a cube is the sum of all \(6\) square faces including top and bottom.

Now, let us find out the total surface area.

We have said that the cube has six square faces.

Let us assume the edge of a cube is \(‘\,x’\) units. One edge of a cube is equal to one side of the square in a face in a cube.

The area of the square is \({x^2}\).

Now, the total surface area of cubes \( = 6 \times {\rm{area}}\,{\rm{of}}\,{\rm{one}}\,{\rm{square}}\)

\(\Rightarrow \)Total Surface Area \( = 6\,\, \times {x^2}\)

\(\Rightarrow \)Total Surface Area \( = 6{x^2}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

Lateral Surface Area of a Cube

The lateral surface area of a cube is the sum of \(4\) square faces excluding the top and bottom.

Let us assume the edge of a cube as \(‘\,x’\) units.One edge of a cube is equal to one side of the square in a face in a cube.

The area of the square is \({x^2}\)

Now, the lateral surface area of cube \( = 4 \times {\rm{area}}\,{\rm{of}}\,{\rm{one}}\,{\rm{square}}\)

\(\Rightarrow \) Lateral Surface Area \({\rm{ =  4 }} \times \,{x^2}\)

\(\Rightarrow \) Lateral Surface Area \( = 4{x^2}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

Surface Area of a Cube Formula

A cube has a lateral surface that excludes the area of the top and bottom faces of a cube and a total surface area that includes all \(6\) faces of a cube.

The formulae for finding the areas of a cube are:

The lateral surface area of a cube \( = 4{x^2}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

Total Surface Area \( = 6{x^2}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

Difference Between the Surface Area of Cube and Cuboid

The cuboid is a solid three-dimensional object which has six rectangular faces. The cuboid is similar to a cube. The number of faces, vertices, and edges of a cube and cuboid is the same. But the main difference between a cube and a cuboid is the length, breadth, and height of a cuboid may vary, whereas, in a cube, the length, breadth and, height of a cube remains the same.

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 \(l\) be the length,\(b\) be the breadth, and \(h\) be the height of a cuboid.

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

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

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

Therefore, the lateral surface area of a cuboid \( = \,2h(l \times b)\;{\rm{sq}}{\rm{.units}}\)

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

So, the total surface area of a cuboid \( = 2\,h\left( {lb + bh + lh} \right)\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

Let us assume the edge of a cube is \(‘\,x’\)

As the cube has all six faces as squares,

The lateral surface area \( = 4 \times {\rm{Area}}\,{\rm{of}}\,{\rm{one}}\,{\rm{square}}\,{\rm{face}} = 4{x^2}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

The total surface area of cube \( = \,{\rm{LSA}}\, + \,{\rm{Area}}\,{\rm{of}}\,{\rm{top}}\,{\rm{and}}\,{\rm{bottom}}\,{\rm{faces}}\)

\({\rm{TSA}} = 4{x^2} + {x^2} + {x^2} = 6{x^2}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)

Surface Area of a Cube Calculator

Q.1. Calculate the cost required to paint a jewellery box that is cubical having an edge length of \(5\,{\rm{cm}}\) if the painting cost is \(₹3\,{\rm{per}}\,{{\rm{m}}^2}\)
Ans: We know that to paint a jewellery box, we need to find the total surface area of it.
\({\rm{TSA}}\) of a cube \( = 6{x^2} \;{\rm{sq}}{\rm{.units}}\)
\( = 6 \times 5 \times 5\,{{\rm{m}}^2}\)
\(= 150\;{{\rm{m}}^2}\;\)
So, the total cost of painting a jewellery box at \(₹3\,{\rm{per}}\,{{\rm{m}}^2} = ₹3 \times 150 =₹ 450\)

Q.2. Find the surface area of a cube with a side of \(10\,{\rm{cm}}\)
Ans: Given \(x = 10\,{\rm{cm}}\)
We know that, the total surface area of a cube \( = 6{x^2}\,{\rm{sq}}{\rm{. units}}\)
So,\({\rm{TSA}}\) of a cube\({\rm{ = }}\,{\rm{6}}\,\, \times \,{{\rm{(10)}}^2}{\rm{c}}{{\rm{m}}^2}\)
\(\Rightarrow {\rm{TSA}}\) of a cube\({\rm{ =  600 \;c}}{{\rm{m}}^2}\)

Q.3.Find the lateral surface area of a cube with a side of \(8\;{\rm{cm}}{\rm{.}}\)
Ans: Given \(x\, = \,8\,{\rm{cm}}\)
We know that, the lateral surface area of a cube \( = 4{x^2}\,{\rm{sq}}{\rm{. units}}\)
So,\({\rm{LSA}}\) of a cube \({\rm{ =  4}} \times \,{8^2}\,{\rm{c}}{{\rm{m}}^2}\)
\( \Rightarrow {\rm{LSA}}\) of a cube \( = 256\,{\rm{c}}{{\rm{m}}^2}\)

Q.4. 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 \(x\,{\rm{unit}}\)
The lateral surface area of a cube \( = 4{x^2}\)
The total surface area of a cube \( = 6{x^2}\)
So, the ratio of \({\rm{LSA}}\) and \({\rm{TSA}}\) of a cube\(= \frac{{{\rm{LSA\;of\;a\;cube}}}}{{{\rm{TSA\;of\;a\;cube}}}} = \frac{{4{x^2}}}{{6{x^2}}}\)
Therefore, the ratio of \({\rm{LSA}}\) and \({\rm{TSA}}\) of a cube \( = \,\frac{4}{6}\, = \,\frac{2}{3}\)

Q.5. A cubical container of side \(6\,{\rm{cm}}\) 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 \(₹20\,{\rm{per}}\,{{\rm{m}}^2}\)
Ans: Given: Length of the side of a cubical container \( = 6\,{\rm{m}}\)
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 that, the total surface area of a cube \( = 6{x^2}\,{\rm{sq}}{\rm{. units}}\)
\({\rm{ =  6}} \times 6 \times 6\,{{\rm{m}}^2}\)
\( = 216\,{{\rm{m}}^2}\)
Also, it is given that the cost of painting \(1\,{{\rm{m}}^2}\) of the cubical container is \(₹20\)
So, the total cost of painting the given cubical container is \(₹20 \times 216 = ₹4320\)
Therefore, the total cost of painting the given cubical container is \(₹4320\)

Summary

From the above article, we learned what a cube is, examples of a cube, its shape, and elements.
Also, we understood the meaning of cube area, surface area of cube formula, surface area of a cube, its types , meaning of surface area of a cube, calculation of the total surface area and the lateral surface area of a cube, and some of the solved examples.

FAQs on Surface Area of a Cube Equation

Students can check the below frequently asked questions on surface area of a cube:

Q1. How do you find the total surface area of a cube?
Ans: We will find the total surface area of a cube with side \(x\) using the formula,
\({\rm{TSA}}\) of a cube \(6{x^2}{\rm\;{sq}}{\rm{.units}}\)

Q2. What is the surface area formula?
Ans: Total surface area is calculated by adding area of all its surfaces. The formula for the total surface area of a cube with side is given by \(6{x^2}\;{\rm{sq}}{\rm{.units}}\)

Q3. What is the surface area of cube and cuboid?
Ans: The lateral surface area of a cuboid \( = \,2h(l\, + \,b)\;{\rm{sq}}{\rm{.units}}\)
The total surface area of a cuboid \({\rm{ =  2(}}lb\, + bh\, + lh)\,{\rm{sq}}{\rm{.units}}\)
The lateral surface area of a cube \( = 4{x^2}\,{\rm{sq}}{\rm{. units}}\)
The total surface area of a cube \( = 6{x^2}\,{\rm{sq}}{\rm{. units}}\)

Q4. What is the surface area of a \({\rm{3}} \times {\rm{3}} \times {\rm{3}}\) cube?
Ans: Total surface area of a cube with side \(x = 6{x^2}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)
The lateral surface area of a cube with side \(x = 4{x^2}\,{\rm{sq}}{\rm{.}}\,{\rm{units}}\)
The volume of a cube with side \(x = {x^3}\,{\rm{cubic}}\,{\rm{units}}\)

Q5. What are the surface area and volume of a cube?
Ans: We know that, the total surface area of a cube \(x = 6{x^2}\;{\rm{sq}}{\rm{.units}}\)
The lateral surface area of a cube with side \(x = 4{x^2}\,{\mkern 1mu} {\rm{sq}}{\rm{.units}}\)
The volume of a cube with side \(x\, = \,{x^3}\,{\rm{cubic}}\,{\rm{units}}\)

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