What is a Cube?
A cube is a three-dimensional solid that is enclosed by six congruent square faces. It is also known as regular hexahedron or square prism. Some real-life examples of cubes are an ice cube, a Rubik’s cube and a regular dice.
A cube consists of six square faces, eight vertices, and twelve edges or sides. All the side lengths of a cube are equal, that is, the length, breadth, and height are of the equal measurement.
The edge of a cube is the common boundary of the faces of a cube.
Formula for the Surface Area and Volume of a Cube
Surface Area of a Cube:
For any 2D shape, area is defined as the region occupied by it in a plane. A cube is a 3D solid, so the area occupied by all the six congruent squares is known as the surface area of the cube. Therefore, to calculate the surface area of a cube, we need to calculate the area of all the six squares, and then find the sum of the area of all the square faces.
Let ‘\(a\)’ be the edge of the cube.
Area of one square face = \(side^2\) = \(a^2\)
A cube having six such square faces:
Area of six square faces = 6 \(\times\) area of one square face
= 6 \(\times\) \(a^2\)
= \(6a^2\)
So, surface area of cube, \(S~=~6a^2\) square units. Where a is the length of its edge.
Lateral Surface Area of a Cube:
Lateral surface area of a cube = Area of all four lateral faces
= 4 \(\times\) area of one square face
= \(4a^2\)
So, lateral surface area of a cube is \(4a^2\) square units.
Volume of a Cube:
The volume of any solid is defined as the total space occupied by it and volume is always measured in cubic units.
Volume of a cube, V = length \(\times\) length \(\times\) length
= \(a~\times~a~\times~a\)
= \(a^3\)
So, the volume of a cube, \(V~=~a^3\) cubic units.
Length of the Diagonal of a Cube
If a is the length of the side or edge,
- Length of Diagonal of Face of the Cube = \(\sqrt{2}\)a
- Length of the Diagonal of Cube = \(\sqrt{3}\) a
Properties of Cube
- All faces of the cube are squares
- All the dimensions of the cube, that is, length, width and height, always remain congruent
- The plane angles of the cube are right angles, that is, equal to 90 degrees
- The edges opposite to each other are parallel
Solved Examples
Example 1: Find the volume of the cubical box whose side length is 15 inches.
Solution:
Volume of a cube = \(a^3\) Write the formula
= \(15^3\) Substitute 15 for \(a\)
= 3375 Find the cube of 15
So, the volume of the box is 3375 cubic inches.
Example 2: Find the side length of a cubical cell whose volume is 343 cubic meters.
Solution:
\(V~=~a^3\) Write the formula
\(343~=~a^3\) Substitute 343 for \(V\)
\(7~=~a\) Apply cube root on both sides
Hence, the side length of the cube is 7 meters.
Example 3: Find the surface area and lateral surface area of a cube whose side length is 5 inches.
Solution:
\(S~=~6a^2\) Write the formula for surface area
\(S~=6~\times~5^2\) Substitute 5 for a
\(S~=6~\times~25\) Find square of 5
\(S~=~150\) Multiply
Lateral surface area of a cube = \(4a^2\)
= \(4~\times~5^2\) Substitute 5 for a
= \(4~\times~25\) Find square of 5
= \(100~in^2\) Multiply
So, the surface area and lateral surface area of the cube are 150 square inches and 100 square inches respectively.
A cube is a three dimensional solid all of whose faces are in the shape of a square. In contrast, a cuboid is a three dimensional solid whose faces are rectangles.
Yes, a cube has identical bases, so it is also a prism.
A cube has six faces in all.
The volume of the cube is calculated by multiplying the edge length thrice.