3D Shapes

What are 3d (Three- dimensional) Shapes?

Geometry basically deals with 2d shapes and 3d shapes. Let us study what are the 3D shapes? Let us draw a picture of a notebook on a piece of paper. What we observe is a plain picture drawn on a paper. It does not occupy any space, but if we keep a real notebook on that piece of paper it occupies some space, and such shapes are called 3D shapes or three-dimensional shapes. 

Plane geometry or two-dimensional geometry deal with the flat figures that can be drawn on a piece of paper like line, curves, polygons, quadrilaterals, etc, while solid geometry or three-dimensional geometry deals with solid shapes or three-dimensional shapes. Examples of three-dimensional shapes are sphere, cylinders, cones, etc. Let us study some of the 3D shapes and properties of 3d shapes.

What are 3D Shapes?

The shapes that occupy space are called three-dimensional shapes. Three - dimensional shapes can also be defined as the solid shapes having three dimensions length, width, and height. A football is an example of the 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 3D shapes all around us like a table, chair, notebook, pen, etc. Here are some of the examples of three-dimensional shapes and properties of 3d shapes.

[Image will be Uploaded Soon]

Some of the Attributes of 3D Shapes are:

  • Faces: The flat figures of the solid figures are called the faces of the 3D figures.

  • Edges: A line segment where the two faces meet are called the edges of the 3D shapes.

  • Vertices: A corner point where the edges of the solid figures meet are called vertices.

The below figure represents faces, edges, and vertices of a cube, an example of the 3D shape.

[Image will be Uploaded Soon]

Let us study some of the common examples of three-dimensional shapes used in geometry and properties of 3d shapes.

Cube

A cube is a three-dimensional shape formed by two-dimensional 6 square faces. The cube has the following properties.

  • All edges and faces are equal.

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

  • The measures of all the angles is 900.

Cuboid

[Image will be Uploaded Soon]

A cuboid is a three- dimensions shape formed by two- dimensional rectangular faces. It is also called a rectangular prism. The cuboid has the following properties

  • Opposite faces and edges are equal.

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

  • All the angles measure 900.

Prism

[Image will be Uploaded Soon]

A solid shape having its base and top as identical polygons and lateral faces as parallelograms are called a prism. Examples: Triangular prism, square prism, Pentagonal prism. The properties of the prism are according to the shapes. 

Some of the properties of a triangular prism are 

It has 6 vertices, 9 edges and 5 faces – 2 triangles and 3 rectangles

Pyramid

[Image will be Uploaded Soon]

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

The most commonly used pyramid is the square pyramid i.e., it has a square base and four triangular side faces.

Some of the properties of a square pyramid are 

It has 5 vertices, 8 edges, and 5 faces

Cylinder

[Image will be Uploaded Soon]

A solid 3D shape which has two circular bases and a curved surface is called a cylinder. Some of the properties of a cylinder are:

  • It has no vertex, 2 edges

  • 2 flat faces which are circles

  • And 1 curved surface

Cone

[Image will be Uploaded Soon]

A solid 3D shape which has one circular base connected with a curved surface and has a single vertex is called a cone. 

Which is perfectly round in shape and Some of the important properties of a cone are

  • It has 1 vertex and 1 edge

  • 1 flat face which is a circle

  • And 1 curved face

Sphere

[Image will be Uploaded Soon]

A solid three-dimensional shape that is perfectly round in shape and all its points on the surface is equidistant from the center point is called a sphere.

The distance between the center and any point on the surface is called the radius of the circle. 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.

Surface Area and Volume of 3D shapes

3D shapes are measured in two terms that are:

  • Surface Area-The area of all the faces of a solid is called the surface area of the object. Surface area is denoted by SA. Total surface area(TSA) is divided into two parts

  • Curved Surface Area is the area of all the curved regions denoted by ‘CSA’.

  • Lateral Surface Area is the area of all the curved regions and all the flat surfaces excluding base areas denoted by ‘LSA’.

Volume

The amount of space that encloses a solid shape is called the volume of the shape. And it is denoted by ‘V’

Here is the formula chart for  surface area and volume of 3D shapes in a tabular format

Formula for Surface Area and Volume of 3d shapes(Solid shapes)

S.No

Name 

Abbreviations used

Lateral /curved Surface Area

Total Surface Area

Volume

1.

Cuboid

h = height, 

l = length  b=breadth

2h(l+b)

6l2

L * b* h 

2.

Cube

a = length of the sides

4a2

6a2

a3

3.

Right Prism

..

Perimeter of Base × Height

Lateral Surface Area + 2(Area of One End)

Area of Base × Height

4.

Right Circular Cylinder

r= radius

h=height

2 (π × r × h)

2πr (r + h)

πr2h

5.

Right pyramid

..

½ (Perimeter of Base × Slant Height)

Lateral Surface Area + Area of the Base

⅓ (Area of the Base) × Height

6.

Right Circular Cone

r = radius

l = length

πrl

πr (l + r)

⅓ (πr2h)

7.

Sphere

r = radius

4πr2

4πr2

4/3πr3

8. 

Hemisphere

r = radius

2πr2

3πr2

⅔ (πr3)


Solved examples:

Example 1:

Find the volume and surface area of a cuboid of l= 10cm, b = 8cm and h = 6 cm.

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

    =10 x 8 x 6

                                       = 480cm2

Surface area = 2 ( lb + lh + bh)

= 2( 10x8 + 10x6 + 8x6)

=2(80 + 60 + 48)

=376cm2

Quiz Time

Find the area of the right-angled triangle whose base is 12cm and hypotenuse 13cm.

  1. 40 cm

  2. 85 cm

  3. 60 cm

  4.  30cm2

The side of a square whose surface area is 600cm is

  1. 10cm

  2. 20 cm

  3. 30 cm

  4. 40 cm

FAQ (Frequently Asked Questions)

1. What is Mensuration in Maths?

Answer: Everywhere around us, there are different 2D and 3D shapes. The branch of mathematics that deals with measurements of different 2D and 3D geometrical shapes. Mensuration formulas are used to measure the length, width, areas, volume, etc. of the different shapes like triangle, circle, sphere, cone, etc.

The Study of mensuration helps us in calculating different measurements of the 2d and 3d shapes. Mensuration is used in day to day life. Some of the examples are:

  • Measurement of fields, for building a house.

  • Measurement of volume of a can, to carry milk.

  • Measurement of the surface area of the walls for the estimation of painting it.

  • Measurements of the volume of food for packaging.