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## Cuboid And Cube

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## Cuboid

What is a cuboid?

• A cuboid is a three-dimensional geometric shape that resembles a rectangular box or a rectangular prism. A cuboid has 3 Pairs of opposite, congruent and parallel rectangular faces, 12 edges, and 8 vertices.
• Note 1: All squares are rectangles.
• Note 2: Cuboid may have one equal pairs of squares.
• Note 3: If All three pairs of faces of a cuboid are squares then it it becomes a Cube.
• Note 4: A cube is a special case of cuboid.

## Parts And Their Alignment Of A Cuboid

### Faces

The flat surfaces of a cuboid are known as its faces.

A cuboid has six faces, and each face is a rectangle.

These faces are arranged such that three pairs of opposite faces are parallel to each other.

The adjacent faces are perpendicular to each other (i.e., the angle between any two touching faces of a cube is right angle, 90°.

• Note 1: All squares are rectangles.
• Note 2: Rectangle may have one or two pairs of squares.
• Note 3: If All three pairs of faces of a rectangle are squares then it it becomes a Cube.
• Note 4: A cube is a special case of cuboid.

### Edges

An edge is a line segment where the two surfaces of a cuboid meet.

There are 12 edges in a cuboid, where three edges meet at each vertex.

All edges form right angles with the adjacent edges and faces.

### Vertices

A vertex is a point where the three edges meet. Vertices is the plural of vertex.

Cuboid has eight vertices.

### Diagonals

Diagonal of a cuboid is a line segment that joins two opposite vertices.

The cuboid has four space diagonals.

Length of the diagonal of cuboid = √(length2 + breadth+ height2) units.

### Symmetry

Cuboids exhibit high symmetry

They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.

## Features of a Cuboid

It is a three-dimensional, Rectangular figure.

It has 6 faces, 12 edges, and 8 vertices.

All 6 faces are rectangles.

Each vertex meets three faces and three edges.

The edges run parallel to those parallel to it.

All angles of a cuboid are right angles.

## Mensuration of Cuboid

### Surface Area of a Cuboid

The total surface area of a cuboid is defined as the area of its surface (Appearing face).

### The Lateral Surface Area of a Cube.

Imagine yourself sitting in a cuboid shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.

That is, the lateral surface area of a cuboid shaped room is the area of its four walls, excluding the ceiling and the floor.

The lateral surface area of the cuboid is the sum of areas of its square faces, excluding the area of the top and the bottom face.

So the lateral surface area of a cube = sum of areas of 4 faces = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height)

Derivation of Total Surface Area of a Cuboid

Since the total surface area of a cuboid (TSA) is the area of its surface.

Total surface area of a cuboid = Lateral Surface Area + Area Of Bottom Surface + Area Of Top Surface

Total surface area of a cuboid = Area Of Front Surface + Area Of Back Surface + Area Of Left Srface + Area Of Right Surface + Area Of Bottom Surface + Area Of Top Surface

Total surface area of a cuboid = Lateral Surface Area 2[Area Of Bottom Surface]

Since Area Of Top Surface = + Area Of Bottom Surface We get, Total surface area of a cuboid = Lateral Surface Area + 2[Area Of Top Surface]

TSA = (Length ✕ Height) + (Length ✕ Height) + (Length ✕ Height) + (Breadth ✕ Height) + (Breadth✕ Height) + (Length ✕ Breadth) + (Length ✕ Breadth)

TSA = 2(Length ✕ Height) + 2(Breadth ✕ Height) + 2(Length ✕ Breadth)

TSA = 2[(Length ✕ Height) + (Breadth ✕ Height) + (Length ✕ Breadth)]

## Volume

The volume of a three-dimensional object can be defined as the space required for it.

Similarly, Volume of a cuboid is defined as the space required for the cuboid or the Space occupied by the cuboid.

The volume of a cuboid can be calculated using the formula, V = lbh, where,

l = length, b = breadth or width, h = height

This formula shows that the volume of a cuboid is directly proportional to its length, breadth and height.

The volume is calculated by multiplying the object’s length, breadth, and height.

Hence the volume of the cube = lbh = lenth ✕ breadth ✕ height

## Cuboids in Our Daily Life

• Cuboids are commonly used in everyday objects, such as boxes, books, and building blocks.
• They are used in architectural and engineering designs for modeling rooms, buildings, and structures.
• In mathematics and geometry, cuboids serve as fundamental examples for teaching and understanding concepts related to three-dimensional shapes.

Similar Shapes:

• A cube is a special type of cuboid where all sides are equal in length, making it a regular hexahedron.

Real-world Examples:

• A shoebox is an example of a cuboid.
• Most refrigerators, ovens, and TV screens have cuboidal shapes.
• Buildings and houses often have cuboidal rooms.

Fun Fact:

• Cuboids are among the simplest and most familiar three-dimensional shapes, making them a fundamental concept in geometry.

Remember that these notes provide an overview of cuboids, and there are more advanced topics and applications related to this shape in various fields of study.

## What is a cube?

A cube is a three-dimensional regular polyhedron characterised by its 6 Identical (Congruent) Squares in which 3 Pairs of them parallel.

## Parts And Their Alignment Of In A Cube

### Faces

The flat surfaces of a cube are known as its faces.

A cube has six faces, and each face is a perfect square. These faces are arranged such that three pairs of faces are parallel to each other.

The adjacent faces are perpendicular to each other (the angle between any two touching faces of a cube is right angle, 90°.

All the edges have the same length.

A cube also has 8 vertices and 12 edges.

### Edges

An edge is a line segment where the two surfaces of a cube meet.

There are twelve edges in a cube, where three edges meet at each vertex.

All edges have equal length and form right angles with the adjacent edges and faces.

### Vertices

A vertex is a point where the three edges meet. Vertices is the plural of vertex.

Cube has eight vertices.

### Diagonals

The cube has four space diagonals that connect opposite vertices, each of which has a length of √3 times the length of an edge.

### Symmetry

Cubes exhibit high symmetry

They have rotational symmetry of order 4, meaning that you can rotate them by 90 degrees about their centre and they will look the same.

## Features of a Cube

It is a three-dimensional, square-shaped figure.

It has 6 faces, 12 edges, and 8 vertices.

All 6 faces are squares with equal area.

All sides have the same length.

Each vertex meets three faces and three edges.

The edges run parallel to those parallel to it.

All angles of a cube are right angles.

## Mensuration of Cube

### Surface Area of a Cube

The total surface area of a cube is defined as the area of its outer surface

Derivation of Total Surface Area of a Cube

Since the total surface area of a cube is the area of its outer surface.

total surface area of a cube = 6 ✕ area of one face.

We know that the cube has six square faces and each of the square faces is of the same size, the total surface area of a cube = 6 ✕ area of one face.

Let the length of each edge is “s”.

Area of one square face = length of edge ✕ length of edge

Area of one square face == s ✕ s = s²

Therefore, the total surface area of the cube = 6s²

The total surface area of the cube will be equal to the sum of all six faces of the cube.

### The Lateral Surface Area of a Cube.

Imagine yourself sitting in a cube shaped room. You can then see the four walls around you. This denotes the lateral surface area of that room.

That is, the lateral surface area of a cube shaped room is the area of its four walls, excluding the ceiling and the floor.

The lateral surface area of the cube is the sum of areas of its square faces, excluding the area of the top and the bottom face.

So the lateral surface area of a cube = sum of areas of 4 faces = 4a²

The Volume of a Cube

## Volume

The volume of a three-dimensional object can be defined as the space required for it.

Similarly, Volume of a cube is defined as the space required for the cube or the Space occupied by the cube.

The volume of a cube can be calculated using the formula V = s3, where “s” represents the length of one side of the cube.

This formula shows that the volume of a cube is directly proportional to the cube of its side length.

The volume is calculated by multiplying the object’s length, breadth, and height. In the case of a cube shape, the length, width, and height are all of the same length. Let us refer to it as “s”.

Hence the volume of the cube is s ✕ s ✕ s = s³

## Cubes in Our Daily Life

We encounter many cubes in our daily life such as Ice cubes, sugar cubes, dice and the building blocks used in games.

Cubes play a fundamental role in the study of geometry and serve as a basis for understanding three-dimensional space and concepts such as volume and surface area.

Also, Cubes have many applications in mathematics, engineering, architecture and art etc.