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A cube is a three-dimensional shape with six square faces that are all congruent to each other. Each face of a cube is perpendicular to the adjacent faces, and all of its edges are the same length. It is a regular polyhedron and is often used in mathematics, geometry, and physics to represent symmetry and spatial relationships.
A cube is sometimes also referred to as a regular hexahedron or as a square prism. It is one of the 5 platonic solids. Some real-life examples of a cube are an ice cube, a Rubik's cube, a regular dice, etc. Let us learn about a cube along with its formulas, a few solved examples, and practice questions here.
|1.||What is a Cube?|
|2.||Properties of Cube|
|5.||FAQs on a Cube|
What is a Cube?
A cube is a 3D solid shape with six square faces and all the sides of a cube are of the same length. It is also known as a regular hexahedron and is one of the five platonic solids. The shape consists of six square faces, eight vertices, and twelve edges. The length, breadth, and height are of the same measurement in a cube since the 3D figure is a square that has all sides of the same length.
In a cube, the faces share a common boundary called the edge which is considered as the bounding line of the edge. The structure is defined with each face being connected to four vertices and four edges, vertex connected with three edges and three faces, and edges are in touch with two faces and two vertices.
Cube Definition in Maths
A cube is a three-dimensional solid figure, having 6 square faces. It is a geometrical shape with 6 equal faces, 8 vertices, and 12 equal edges. Some of the real-life cube examples are playing dice, ice cubes, Rubik's cube, etc that we see around us.
Properties of Cube
A cube is considered a special kind of square prism since all the faces are in the shape of a square and are platonic solid. There are many different properties of a cube just like any other 3D or 2D shape. The properties are:
- A cube has 12 edges, 6 faces, and 8 vertices.
- All the faces of a cube are shaped as a square hence the length, breadth, and height are the same.
- The angles between any two faces or surfaces are 90°.
- The opposite planes or faces in a cube are parallel to each other.
- The opposite edges in a cube are parallel to each other.
- Each of the faces in a cube meets the four other faces.
- Each of the vertices in a cube meets the three faces and three edges.
A cube net is a two-dimensional representation of a cube that shows all six faces of the cube laid out flat so that the net can be cut out and folded into a three-dimensional cube. Through the net of the cube, we can clearly see the six faces i.e. the six square faces that combine together at the edges to form a cube. Here is an image for your reference:
The cube formulas helps us to find the surface area, diagonals, and volume of a cube. Here is the list of all cube formulas:
- LSA (Lateral Surface Area) of cube = 4a2
- TSA (Total Surface Area) of cube = 6a2
- Volume of cube = a3 (or) (√3×d3)/9
- Diagonal of face of cube = a√2
- Main Diagonal of cube = a√3
In all these formulas, 'a' represents the length of each edge and 'd' represents the main diagonal of the cube. Let us discuss the different formulas of a cube in detail.
Surface Area of a Cube
There are two types of surface areas of a cube - Lateral Surface Area (LSA) and Total Surface Area (TSA)
Lateral Surface Area of a Cube
The lateral area of a cube is the sum of areas of all side faces of the cube. There are 4 side faces so the sum of areas of all 4 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.
LSA of a Cube = 4a2
where a is the side length. For more information, you can check this interesting article on lateral area of a cube formula.
Total Surface Area of a Cube
The total surface area of the cube is the sum of the area of the base and the area of the vertical surfaces of the cube. Since all the faces of the cube are made up of squares of the same dimensions, the total surface area of the cube will be the surface area of one face added six times to itself. It is measured as the "number of square units" (square centimeters, square inches, square feet, etc.). Therefore, the formula to find the surface area of a cube is:
Total Surface Area (TSA) of a Cube = 6a2
where a is the side length. For more information, you can check out this interesting article on surface area of cube.
Volume of a Cube
The volume of a cube is the space occupied by the cube. The volume of a cube can be found by finding the cube of the side length of the cube. To determine the volume of a cube, there are different formulas based on different parameters. It can be calculated using the side length or the measure of the cube's diagonal and it is expressed in cubic units of length. Hence, the two different formulas to find the volume of the cube are:
- The Volume of a Cube (based on side length) = a3 where a is the length of the side of a cube
- The volume of a Cube (based on diagonal) = (√3×d3)/9 where d is the length of the diagonal of a cube
You can read more on the volume formula by reading this interesting article on volume of cube.
Diagonal of a Cube
The diagonal of a cube is a line segment that joins two opposite vertices of a cube. The length of the diagonal of a cube can be determined using the diagonal of a cube formula. It helps in finding the length of the face diagonals and the main diagonals. Each face diagonal forms the hypotenuse of the right-angled triangle formed. A cube has six faces (square-shaped). On each face, there are two diagonals joining the non-adjacent vertices. Therefore, we have twelve face diagonals and four main diagonals that connect the opposite vertices of the cube. The diagonal of a cube formula to calculate the length of a face diagonal and the main body diagonal of a cube is given as,
- Length of face diagonal of cube = √2a units, where a = Length of each side of a cube
- Length of main diagonal of a cube = √3a units, where a = Length of each side of a cube
Let us have a look at a few solved examples on the cube and its properties for a better understanding.
Listed below are a few topics that are related to a cube.
Example 1: What is the amount of water stored in one ice cube of side length 5 cm?
Length of the ice cube = 5 cm
Amount of water stored in the ice cube = Volume of the cube
Therefore, the volume of the ice-cube = 5 × 5 × 5 cm3
= 125 in3
Answer: The amount of water in the ice is 125 cm3.
Example 2: Find the total surface area of the cube if the length of the side of the cube is 25 in.
Length of the side of the cube, a = 25 in
Using the formula for the area of the cube, which is: A = 6a2
A = 6 × 25 × 25
A = 3750
Answer: The surface area of the cube is 3750 square inches.
Example 3: Find the volume of a Rubik's cube of length 6 in.
To find the volume of a Rubik's cube:
The length of the side of the cube= 6 in. (given)
Using the cube formula,
volume = s × s × s = s3
Put the values,
volume = 6 × 6 × 6 = 63 = 216
Answer: The volume of a Rubik's cube is 216 cubic inches.
FAQs on Cube
What is the Meaning of Cube in Geometry?
In geometry, a cube is a three-dimensional geometric shape with six congruent square faces. A perfect real-life example of a cube is an ice cube. It is one of the five platonic solids and is also known as a regular hexahedron.
What are the Two Main Properties of a Cube?
A cube is a three-dimensional shape with many geometrical properties. The two major properties are listed below.
- A cube has 12 edges, 6 faces, and 8 vertices.
- All the faces of a cube are square-shaped.
Tell All About Cube.
A cube is a three-dimensional shape with 6 congruent squares as its faces where every two adjacent faces are perpendicular to each other. It has 8 vertices, 12 edges, and 6 faces. If 'a' is the side length of the cube then, the formulas of cube are:
- volume = a3
- total surface area = 6a2
- lateral surface area = 4a2
- diagonal of cube = √3a
Why is a Cube Called a Regular Hexahedron?
A regular hexahedron is a three-dimensional object with 6 congruent faces. Thus, a cube is called a regular hexahedron.
What is the Formula For the Lateral Area of a Cube?
The lateral area of a cube can be calculated given its edge length. The lateral area of a cube of edge length 'x' is 4x2 square units.
What are the Differences Between a Cube and a Cuboid?
Here are the differences between cube and cuboid.
|Shape||All faces are squares in a cube.||Cuboid is a rectangular shape with six rectangular faces.|
|Lengths||All lengths (a) are same.||All lengths (l, b, h) are not same.|
|Surface Area||6a2||2(lb + bh + hl)|
|Diagonals||a√3||√(l2 + b2 + h2)|
|Symmetry||Has rotational symmetry of order 4.||No rotational symmetry|
How Do You Find the Lateral Area of a Cube?
The lateral area of a cube of edge length 'x' can be obtained by adding the areas of 4 side faces. Thus, the lateral area of the cube = x2 + x2 + x2 + x2 = 4x2.
What is the Difference Between the Surface Area and Lateral Area of a Cube?
The surface area (or) total surface area (TSA) of a cube is the sum of areas of all faces whereas the lateral surface area (LSA) is only the sum of the 4 side faces of the cube. If 'x' is the edge length of the cube, then
- Total Surface Area (TSA) = 6x2
- Lateral Surface Area (LSA) = 4x2
What is Surface Area and Area?
Usually, the term "area" is used to represent the space enclosed by a two-dimensional object. The "surface area" is used to represent the sum of the areas of all faces of a three-dimensional object.
What is the Volume of Cube Formula?
The volume of a cube can be calculated given the side length. The volume of a cube is a3, where a is the length of the side of the cube.
What is the Formula to Find the Area of the Base of a Cube?
The formula to find the area of the base of a cube is a2, where a is the length of the side of the cube.
What Do the 5 Platonic Solids Represent?