Solids

This section will introduce the students to one of the most important aspects of Maths, i.e, Solids. Solids are very important from the perspective of Maths and measurement,  and the analysis of their volume, surface area, and total area is very much useful in day-to-day life. In our day-to-day life, we come across a number of different types of solids.  A room may be in the shape of a cube or cuboid, the cricket ball is in the shape of a sphere, the birthday hat or ice cream is in the shape of a cone, water tanks on the rooftops are in the shape of a cylinder, the gift boxes are in the shape of cubes or cuboids, etc. Similarly, different things that we use in our day-to-day life are in the shape of different solid figures, a combination of two or more solid figures of the same types or different types, or have a slightly modified shape than the actual solid. For example, a water bottle can be a combination of two different cylinders, one of larger radius and another of a smaller radius, a pen is a combination of two cylinders, a refill in the shape of a cylinder with a smaller radius and the body is also in the shape of a cylinder with a larger radius, a dome is formed by the combination of two types solid shape, i.e., a hemisphere and a cylinder/cone. The hemisphere is mounted on a cylinder, or it may be mounted on a cone.  

The discussion will be on different types of solids, like a cylinder, prism, sphere, cube, rectangle, etc., volume, surface area, and total area of the solids, properties of the sides and diagonals of the solids, the characteristic of their interior and exterior angles, and so on. Students will also be introduced to a formula list that will summarise the formulae for volume and different types of areas of the respective solids. 

The formula list provides a relationship between the different parameters of the solid figure and its three-dimensional aspects of volume and area. Using the list makes the memorization of the formulae for different types of solids very easy and helps in creating links between different formulae. Some solved examples are also provided for better understanding and comprehension of the students.    

Geometry basically deals with two types of shapes: plane shapes and solid shapes. Let us study what solids are in maths. We have observed that a plain picture can be drawn on paper, it does not occupy any space. But if we keep a real picture on that piece of paper, it occupies some space, and such shapes are called solid shapes or solids or three-dimensional shapes. 

Solid geometric shapes or three-dimensional geometry deals with solids or three-dimensional shapes. Solids examples are cubes, cuboids, spheres, cylinders, cones, etc. Let us study types of solid shapes and solid shapes properties.

What are Solids in Maths?

The shapes that occupy space are called three-dimensional shapes or solids. Solid shapes can also be defined as figures having three dimensions - length, width, and height. A ball is an example of a sphere, which is a three-dimensional figure, while a circle drawn on a piece of paper is a two-dimensional figure. Similarly, we have many solid shapes all around us, like a table, chair, notebook, pen, etc. Here are some of the solid examples and solid shapes properties.

(Images will be uploaded soon)

Some of the Attributes of Solid Geometric Shapes are

• Faces: The single flat figures of the solid figures are called the faces of the solid figures. For example, it may be square or rectangular, or any polygon.

• Edges: Line segments between two faces where the two faces meet are called the edges of the solid shapes.

• Vertices: A corner point where the edges of the solid figures meet are called vertices. Generally, three faces meet at a single vertex. The plural form of the vertex is vertices.

The below figure represents the faces, edges, and vertices of a pyramid, an example of a solid shape.

(Images will be uploaded soon)

Types of Solid Shapes

Let us study some of the types of solids in maths, common solids examples used in geometry, and solid shapes properties.

The different types of solids can be found to be three-dimensional extensions of different two-dimensional shapes, like a cube can be seen as a three-dimensional extension of a square. Similarly, a cylinder can be seen as a three-dimensional extension of a circle. Analyzing the solids this way will help in understanding the formulae and properties of the different types of solids easily. 

Different types of solid shapes are explained in detail.

Cube

(Images will be uploaded soon)

A cube is a solid formed by two-dimensional 6 square faces. It can be seen as an extension of the square in three dimensions. Its properties are as follows:

• All its edges and faces are equal.

• 8 vertices

• 12 edges

• 6 faces

• The measures of all the angles are 900.

Cuboid

(Images will be uploaded soon)

A cuboid is a solid formed by two-dimensional rectangular faces, also called a rectangular prism. It has the following properties:

• Opposite faces and edges are equal.

• 8 vertices 

• 12 edges 

• 6 faces

• All the angles measure 900.

Prism

(Images will be uploaded soon)

A solid which has its base and top as identical polygons and lateral faces as parallelograms is called a prism. Examples: triangular prism, square prism, pentagonal prism.

Some of the properties of a triangular prism are as follows: 

• 6 vertices 

• 9 edges 

• 5 faces – 2 triangles and 3 rectangles

Pyramid

(Images will be uploaded soon)

A solid that has its base as any polygon and side face as triangles with a common vertex is called a pyramid. 

For example, the square pyramid has a square base and four triangular side faces.

Some of the properties of a square pyramid are as follows:

•  5 vertices 

• 8 edges 

• 5 faces

Cylinder

(Images will be uploaded soon)

A solid shape that has two circular bases and a curved surface is called a cylinder. Some of the properties of a cylinder are as follows:

• no vertex 

• 2 edges

• 2 flat faces, which are circles

• 1 curved surface

Cone

(Images will be uploaded soon)

A solid shape that has one circular base connected with a curved surface and has a single vertex is called a cone. 

Some of the important properties of a cone are as follows:

• 1 vertex

• 1 edge

• 1 single flat face, which is a circle

• 1 single curved face

Sphere

(Images will be uploaded soon)

A solid that is perfectly round in shape is called a sphere.

Some of the important properties of the sphere are: 

• It has no vertex.

• It has no edges.

• It has no flat faces.

• It has only 1 curved face.

Solid Formulae

Here is the tabular list of the common solid formula, which will be easier to access.

Solid Formula List

Memorize these solids equations to solve the solid maths questions.

It contains different solids equations. These solid equations are widely used in geometric problems. The formula list contains formulae for volume, surface area, and total area of different types of solids. 

The formula list will prove to be very useful for solving different types of problems on volume, surface area, and total area of different types of solids.

Solved Examples

Example 1: Find the volume and surface area of a cuboid of l= 5 cm, b = 6 cm, and h = 7 cm.

Solution:

Given that l= 5 cm, b = 6 cm, and h = 7 cm,

 We have Volume of cuboid = V = l x b x h 

=5 x 6 x 7

= 210\[cm^{3}\]

Surface area = 2 ( lb + lh + bh)

= 2( 5x6 + 6x7 + 5x7)

=2(30 + 42 + 35)

=2(107)

= 142\[cm^{2}\]

Example 2:  Find the volume of the cylindrical water tank having a radius of base = 30 cm and height of the tank as 50 cm.

Solution: Given the radius of the cylinder r = 30 cm,

The Height of the cylinder is h= 50 cm.

The volume of the cylinder = \[\pi r^{2}h\]

= (22/7)× 30×30×50

=141428.57 \[cm^{3}\]

Example 3: Find the volume of the sphere of radius 28 cm.

Solution:

Given radius of the sphere = r = 28 cm,

Volume of sphere = \[ \frac{4}{3} \pi r^{3}h\]

= (4/3) × (22/7) × 28 × 28 × 28

= 4 × 22 × 28 × 28 x 4

= 275,968\[cm^{3}\]