Solids - Types of Solids, Formula List and Solved Examples

Geometry basically deals with two types of shapes plain shapes and solid shapes. Let us study what are solids in maths? We have observed that a  plain picture can be drawn on a 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 the 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 solids examples and solid shapes properties.


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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: A line segment 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 faces, edges, and vertices of a pyramid, an example of a solid shape.

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Types of Solid Shapes

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

Different types of solid shapes are explained in detail.


Cube

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A cube is a solid formed by two-dimensional 6 square faces.

It has,

  • All its edges and faces equal.

  • I8 vertices,

  • 12 edges,

  • 6 faces.

  • The measures of all the angles are 900.


Cuboid

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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, and 

  • 6 faces.

  • All the angles measure 900.


Prism

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

Some of the properties of a triangular prism are as follows. It has 

  • 6 vertices, 

  • 9 edges and 

  • 5 faces – 2 triangles and 3 rectangles


Pyramid

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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 i.e., it has a square base and four triangular side faces.

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

It has

  •  5 vertices, 

  • 8 edges, and 

  • 5 faces


Cylinder

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

It has 

  • no vertex, 

  • 2 edges

  • 2 flat faces which are circles

  • And 1 curved surface


Cone

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A solid shape which 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.

It has 

  • 1 vertex and 

  • 1 edge

  • 1 single flat face which is a circle

  • And 1 single curved face


Sphere

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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

  •  And no edges.

  • It has no flat faces.

  • It has only 1 curved face.


Solid Formulas

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

It contains different solids equations. These solid equations are widely used in geometric problems.


Solid Formula List


S.No

Name 

Figure

Abbreviations used

Lateral /curved surface area

Total surface area

Volume

1.

Cuboid

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H = height, 

l = length  b=breadth

2h(l+b)

6l2

L * b* h 

2.

Cube

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a = length of the sides

4a2

6a2

a3

3.

Right Prism

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..

Perimeter of Base × Height

Lateral Surface Area + 2(Area of One End)

Area of Base × Height

4.

Right Circular Cylinder

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r= radius

h=height

2 (π × r × h)

2πr (r + h)

πr2h

5.

Right pyramid

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..

½ (Perimeter of Base × Slant Height)

Lateral Surface Area + Area of the Base

⅓ (Area of the Base) × Height

6.

Right Circular Cone

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r = radius

l = length

πrl

πr (l + r)

⅓ (πr2h)

7.

Sphere

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r = radius

4πr2

4πr2

4/3πr3

8. 

Hemisphere

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r = radius

2πr2

3πr2

⅔ (πr3)


Memorize these solids equations to solve the solid maths question.


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

= 210cm3

Surface area = 2 ( lb + lh + bh)

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

=2(30 + 42 + 35)

=2(107)

= 142cm2


Example 2. Find the volume of the sphere of radius 28 cm.

Solution:

Given radius of the sphere = r = 28 cm

Volume of sphere = 4/3 πr3

= (4/3) × (22/7) × 28 × 28 × 28

= 4 × 22 × 28 × 28 x 4

= 275,968cm3

FAQ (Frequently Asked Questions)

1. What is the Difference Between Area and Surface Area?

Answer :

  • The area is the measurement of the space occupied by any two-dimensional solid geometric shapes. Whereas the surface area is the sum of areas of all the faces of the three-dimensional or a solid.

  • Plane 2D figures represent the area. Example: circles, rectangles, and triangles. While solid 3D figures represent the surface area. Examples: cylinder, prisms, pyramids, and cones.

  • When we have to calculate area all we have to do is concentrate on the area of one figure. But in the surface area, when we have to calculate surface area, we have to work out on the area of all the faces.

2. What are Solids in Maths?

Answer: The shapes that occupy space are called three-dimensional shapes or solids, also termed as 3D shapes. Solid shapes are defined three dimensions because they have three dimensions, length, width, and height. A ball is an example 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.