How to Identify and Solve Problems Involving Cube Shapes
In Euclidean Geometry, a cube is a regular solid shape with six square faces, twelve edges, and eight vertices. A cube is also said to be a regular hexahedron or square prism as all its faces are square in shape. The dimensions i.e. the length, breadth, and height of a cube are equal. An ice cube, Rubik’s cube, and dice, etc. are some of the real life examples of a cube. Read on to know more about geometric cube in detail.
Cube Definition in Maths
A cube is defined as a three dimensional solid object that has six identical square faces. The six identical square faces of a cube meet each other at right angles.
The cube, also known as regular hexahedron, is one of the five platonic solid objects. It has 6 faces, 12 vertices, and 8 edges.
Cube Shape Examples
Here are some of the real-life examples of a cube which are mostly observed in everyday life.
Ice Cubes
Dice
Sugar Cube
Rubik Cube
Cube Building Blocks
Chocolate Cube
Cube Diagram
The following cube diagram shows 6 square faces, 12 vertices, and 8 edges.
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Parts of Cube Shape Diagram
The description of the cube shape parts are as follows:
Face: A cube has six flat faces, or surfaces. The face of a cube is also referred to as facets or sides. Each cube faces are square in shape. Among six faces of a cube, each and every face has four equal sides. The sides of each cube face are known as edges.
Edge: The edge of a cube is a line segment joining two vertices. There are 12 edges of a cube. All the 12 edges of a cube are equal in length because the all six faces of a cube are square in shape.
Vertex: A point where three edges meet is known as vertex. There are a total of 8 vertices in a cube.
Properties of Cube
A cube has 6 faces , 8 vertices, and 12 edges.
As all the faces of a cube are square in shape, therefore, the length, breadth, and height of a cube are equal.
The angle between any two sides of a cube is 90 degrees.
Each face of a cube meets the other four adjacent faces.
The opposite sides of a cube are equal and parallel to each other.
The vertex of a cube is a point where two or more edges meet.
Cube Formulas
The following cube formulas help you to calculate the volume and surface area of the cube.
Volume of Cube
The volume of a cube is defined as the total number of cubic units required to fill the given cube completely.
Units of Volume of Cube includes:
Metric: Cubic meters (m), cubic centimeters (cm), liters, etc.
US Standard Unit: Cubic inch, Cubic foot, pints, gallons, etc.
The formula for finding the volume of cube is given as:
The volume of the cube can also be found using the diagonals of a cube. The volume of cube formula using the diagonals of a cube is given as:
Surface Area of Cube
Here, we will learn how to find total surface and lateral surface area of cube using formulas:
Total Surface Area of Cube
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The total surface area of a cube is the sum of the areas of its six faces or sides.
Look at the cube given above. Each of the six faces of the cube is square in shape.
As faces are square, therefore, area of each face = a2
Total surface area (TSA) of a given cube = a2 + a2 + a2 + a2 + a2 + a2
Therefore, TSA = 6a2
Lateral Surface Area of Cube
The lateral surface area of a cube is the sum of the areas of all its faces excluding the top and bottom.
Therefore, the lateral surface area of a cube is given as 4a2
Solved Examples
1. Find the total surface area of the cube of side length 5 cm.
Solution:
Total Surface Area (TSA) = 6a2
Given side length 5 cm, the total surface area of the cube is calculated as
TSA = 6(5)2
TSA = 6 52
TSA = 150 cm2
2. Find the volume of the cube with side length 9 cm.
Solution:
As we know,
Volume of cube = (side)
Here, side length = 9 cm
= 9 9 9
= 729 cm3
Therefore, the volume of cube = 729 cm3
3. Find the lateral surface area of the cube if the length of the side of the cube is 7 cm.
Solution:
Lateral Surface Area (TSA) = 4a2
Given side length 5 cm, the total surface area of the cube is calculated as
LSA = 4(7)2
LSA = 4 72
LSA = 196 cm2
Cube Related Facts
FAQs on Cube Shape in Maths: Meaning, Formulas, Examples
1. What are the main properties of a cube in geometry?
A cube is a three-dimensional solid object defined by several key properties:
It has 6 identical square faces.
It has 12 edges of equal length.
It has 8 vertices (corners).
The angles between any two adjacent faces are 90 degrees.
2. What are the formulas for calculating the volume and surface area of a cube?
The main formulas for a cube with side length 'a' are:
Volume (V): The space occupied by the cube, calculated as V = a³.
Lateral Surface Area (LSA): The area of the four side faces, calculated as LSA = 4a².
Total Surface Area (TSA): The total area of all six faces, calculated as TSA = 6a².
3. What is the main difference between a cube and a cuboid?
The main difference lies in their faces and edge lengths. A cube has 6 identical square faces, meaning all its edges (length, width, and height) are equal. A cuboid has 6 rectangular faces, where only opposite faces are equal, and its length, width, and height can be different.
4. Why is a cube also called a regular hexahedron?
The name 'regular hexahedron' explains its geometric nature. 'Hexa' means six and 'hedron' means face, so a hexahedron is a polyhedron with six faces. The term 'regular' is used because all its faces are identical regular polygons (squares), and the same number of faces meet at each vertex.
5. What is the 'net' of a cube, and how many unique nets can it have?
The net of a cube is a two-dimensional pattern of connected squares that can be folded along the edges to form a three-dimensional cube without any overlaps. While many arrangements are possible, there are exactly 11 unique nets that can form a cube.
6. How do you find the length of the longest diagonal inside a cube?
The longest diagonal of a cube, also known as the space diagonal, connects two opposite vertices through the interior of the cube. If the side length of the cube is 'a', the length of this diagonal can be calculated using the formula: d = a√3. This is an application of the Pythagorean theorem in three dimensions.
7. Is every cube a cuboid? Explain why or why not.
Yes, every cube is a special type of cuboid. A cuboid is defined as a 3D shape with six rectangular faces. Since a square is a special type of rectangle where all four sides are equal, a cube (with six square faces) fits the definition of a cuboid. It is a cuboid where the length, width, and height are all equal.
8. Can you give some real-world examples of objects shaped like a cube?
Many common objects are shaped like cubes, which makes it an important shape to understand. Examples include:
A standard six-sided die
A Rubik's Cube
Sugar cubes
Ice cubes made in a standard tray
Some types of gift boxes
9. How does doubling the side length of a cube affect its surface area and volume?
Doubling the side length has a significant, but different, impact on surface area and volume. If the side length is doubled from 'a' to '2a':
The surface area increases by a factor of four (since TSA = 6(2a)² = 24a² = 4 × 6a²).
The volume increases by a factor of eight (since V = (2a)³ = 8a³).
This shows the non-linear relationship between a cube's dimensions and its properties.
