Table of Contents
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Introduction to 3Dimensional Shapes
3D shapes are nothing but solids that have three dimensions  length, breadth, and height.
The D in 3D stands for Dimensional.
These 3D shapes occupy space and are found in our daytoday life.
We touch, feel and use them.
Some examples of 3D shapes are books, balls, ice cream cones, jugs, donuts etc.
3Dimensional Shapes in Mathematics
In mathematics, we study about 3dimensional objects in the concept of solids and try to apply them in real life.
We will now learn about each 3D shape in detail.
3D shapes are classified into several categories.
Some of them have curved surfaces, pyramids, or prisms.
Let us first learn about the 3dimensional shapes with curved surfaces.
Sphere
 It is shaped like a ball and is perfectly symmetrical.
 Every point on the sphere is at an equal distance from the centre.
 It has one face, no edges, and no vertices.
Cylinder
 It has one curved side.
 The shape stays the same from the base to the top.
 It is a threedimensional object with two identical ends that are either circular or oval.
Cone
 A cone has a circular or oval base with an apex (vertex).
 A cone is a rotated triangle.
 Based on how the apex is aligned to the centre of the base, a right cone or an oblique cone is formed.
Torus
 A torus is a regular ring, shaped like a tyre or donut.
 It is formed by revolving a smaller circle around a larger circle.
 It has no edges or vertices.
Now let us learn about the three dimensional shapes called pyramids.
Pyramid
 A pyramid is a polyhedron with a polygon base and an apex with straight lines.
 Based on its apex alignment with the centre of the base, they can be classified into regular and oblique pyramids.
 A pyramid with a:
 Triangular base is called a Tetrahedron
 Quadrilateral base is called a square pyramid
 Pentagon base is called a pentagonal pyramid
 Regular hexagon base is called a hexagonal pyramid
Let's learn about the threedimensional shapes  Prisms.
Prisms
 Prisms are solids with identical polygon ends and flat parallelogram sides.
 It has the same crosssection all along its length.
 The different types of prisms are  triangular prisms, square prisms, pentagonal prisms, hexagonal prisms, etc.
 Prisms are also broadly classified into regular prisms and oblique prisms.
Next, let's learn about 3dimensional shapes with regular polyhedrons (Platonic Solids).
Polyhedrons / Platonic solids
 There are five polyhedrons.
 Tetrahedron with four equilateraltriangular faces
 Octahedron with eight equilateraltriangular faces
 Dodecahedron with twelve pentagon faces
 Icosahedron with twenty equilateraltriangular faces
 Cube with six square faces
 They have identical faces of regular polygons.
 Song to remember 3D shapes:
3D shapes are fat, not flat.
Find a cone in a birthday hat!
You see a sphere in a basketball,
And a cuboid in a building so tall!
You see a cube in the dice you roll, And a cylinder in a shiny flag pole!

Make these shapes using a plain sheet of paper. A visualization of these complex shapes would help you understand them better.

Moving your fingers along geometric solids will help you understand the concept of faces, edges, and vertices.
Surface Area and Volume of 3D Shapes
Have you ever wrapped a birthday gift?
When you wrap a gift, you cover the surface area of a cube/cuboid which is a 3dimensional object.
Similarly, filling a glass by pouring milk into the glass means that you have filled the volume of the glass (cylinder).
We will learn about the surface area and volume of 3D shapes in detail, but before we do that, let's look at simulation.
Move the slider so that you can see the folding and unfolding of wrapping paper for a cuboidshaped box.
Surface area refers to the area of all the outside surfaces of a threedimensional object. It is denoted by "SA".
There are three common types of surface areas in 3dimensional shapes:
 Curved surface area (CSA)
 Lateral Surface area (LSA)
 Total surface area (TSA)
Volume refers to all the space inside a threedimensional object. It is denoted by "V".
Surface Area and Volume of 3D Shapes with Curved Surfaces
3Dimensional Shapes  Pyramids
There are three types of pyramids namely triangular pyramids, square pyramids and pentagonal pyramids.
However, the most common pyramids are square or quadrilateral pyramids.
Surface Area of a Pyramid:
\(Base\: area + \frac{1}{2} \times (perimeter) \times(slant\: height)\) 
Volume of a Pyramid:
\(\frac{1}{3} \times (Base\: area) \times( height)\) 
3Dimensional shapes  Prisms:
Prisms usually have flat faces and their crosssection is the same across its length.
They are found in regular and irregular shapes.
Based on the alignment of the apex to the centre of the base, they are classified as right prisms or oblique prisms.
Surface Area of a Prism:
\(2\times (Base\: area)+ Base\: perimeter \times length\) 
Volume of a Prism:
\(Base\: area \times length\) 
Properties of 3D Shapes
Sphere:
{Where, Radius (r)}
Properties  Surface Area  Volume 


\(4\pi r^2\)  \(\frac{4}{3}\pi r^3\) 
Cylinder:
{Where, Radius (r), Height(h)}
Properties  Surface Area  Volume 


\(2\pi r(r+h)\)  \(\pi r^2h\) 
Cone:
{Where, Radius (r)}
Properties  Surface Area  Volume 


\(\pi r(r+s)\)  \( \sqrt{3}\pi r^2h\) 
Cube:
{Where edge length (e L)}
Properties  Surface Area  Volume 


\(6 \times (e\: L)^2\)  \((e\: L)^3\) 
Prism:
{Where base area (B A), Perimeter (P), Length (L)}
Properties  Surface Area  Volume 


\(2\times (B\: A)\)+ \(B\:P \times L\)  \(B\: A\times L\) 
Pyramid:
{Where base area (B A), Perimeter (P), Slant Height (S H)
Properties  Surface Area  Volume 


\(B\:A\) + \(\frac{1}{2} \times P \times(S\: H)\) 
\(\frac{1}{3} \times B\: A \times H\) 
Faces, Edges, and Vertices
As mentioned before, 3D shapes and objects are different from 2D shapes and objects because of the presence of the three dimensions  length, breadth, and height.
As a result of these three dimensions, these objects have faces, edges, and vertices.
Let's understand these three in detail.
Faces of 3D Shapes
 A face refers to any single flat surface of a solid object.
 3D shapes can have more than one face.
Edges of 3D Shapes
 An edge is a line segment on the boundary joining one vertex (corner point) to another.
 They serve as the junction of two faces.
Vertices of 3D Shapes
 A point where two or more lines meet is called a vertex.
 It is a corner.
 The point of intersection of edges denotes the vertices.
For example:
3 D shapes  Faces  Edges  Vertices 

Sphere 
1 
0  0 
Cylinder 
2 
2  0 
Cone 
2 
2  1 
Cube 
6 
12  8 
Rectangular Prism 
6 
12  8 
Triangular Prism 
5  9  6 
Pentagonal Prism 
7  15  10 
Hexagonal Prism 
8  18  12 
Square Pyramid 
5  8  5 
Triangular Pyramid 
4 
6  6 
Pentagonal Pyramid 
6  10  6 
Hexagonal Pyramid 
7  12  7 
Examples of 3D Shapes
To summarize the difference between 2D and 3D objects, we can say that 2dimensional objects are flat while 3dimensional objects have depth in them.
Look at the table below to distinguish between 2D and 3D shapes and objects.
2D Shapes  3D Objects 

Circle 
 Sphere  Ball 
Rectangle 
Rectangular prism  Book 
Square 
Cube  Rubik's cube 
Triangle 
Cone  Ice cream cone 
Quadrilateral 
Pyramid  Pyramids in Egypt 
Rectangle and circle 
Cylinder  Bucket 
How to make 3D Shapes for Math Projects
We can make 3D shapes for our math projects using geometrical nets.
A 2dimensional shape that can be folded to form a 3dimensional object is known as a geometrical net.
A solid may have different nets.
In simple words, the net is an unfolded form of a 3D figure.
Let's practice making 3D shapes using the simulations shown below.
Move the slider to see how the nets shape into threedimensional figures.
The following simulation shows 3D shapes with curved surfaces.
Move the slider and the screen to explore the view from different angles.
The next simulation shows how to create 3D shapes which have the properties of a regular polyhedron.
You can create your own nets with a piece of paper by using the templates given beneath each shape.
 Threedimensional objects have 3 dimensions namely length, breadth and height.
 3D shapes have faces, edges and vertices.
 Surface area is classified into:
 Curved surface area
 Lateral surface area
 Total surface area
 Learning about 3D solids will help us in our day to day life as most of our activities revolve and depend on them.
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Solved Examples on 3D Shapes
Example 1 
A mason wants to build a 3D sphere using cement.
He wants to know the amount of cement required to construct the sphere of radius 10 cm.
Find the volume of the sphere using the given radius.
Solution:
Given,
The radius of the sphere (r) = 10 cm
We know the formula for the volume of a sphere:
\(\begin{align}v=\frac{4}{3}\pi\ r^3 \end{align}\) 
The volume of the cement sphere \(v=\frac{4}{3}\pi\ r^3\)
Substituting the value of the radius in the above formula, we get:
\(v=\frac{4}{3}\pi\ r^3\)
\(v=\frac{4}{3}\pi (10)^3\)
\(v= 4188.8 \: cm^3\)
\(\therefore\) volume of the cement sphere is \(4188.8\: cm^3\) 
Example 2 
Find the surface area of a cuboid of length 3 cm, breadth 4 cm, and height 5 cm.
Solution:
Given that,
Length of the cuboid = 3 cm
Breadth of the cuboid = 4 cm
Height of the cuboid = 5 cm
Surface area of the cuboid is
\(\begin{align}2 \times (lb + bh + lh) \end{align}\) 
\(= 2 \times (lb + bh + lh)\)
\(= 2(3\times 4 + 4 \times 5+ 3\times 5\))
\( = 2(12+20+15\))
\(=2(47)\)
\(=94 \: cm^2\)
\(\therefore\) Surface Area of cuboid = \(94\: cm^2\) 
Example 3 
Bhavya wants to drink milk in a glass which is in the shape of a cylinder.
The height of the glass is 15 cm and the radius of the base is 3 cm.
What is the quantity (volume) of milk that she requires to fill the glass?
Solution:
Given that,
Height of the glass = 15 cm
Radius of the glass base = 3 cm
To find the volume of the glass, we need to use the formula for the volume of a cylinder, which is
\(\begin{align}\pi r^2h \end{align}\) 
The volume of the glass, V = \(\pi r^2h\)
\(V= \pi (3)^2 \times 15\)
\(V= \pi (135)\)
\(V= 424.11\:cm^3\)
Therefore, she needs roughly 425 \(cm^3\) of milk to fill her glass.
\(Volume =424.11\: cm^3\) 
Example 4 
Identify the regular polyhedron from the images shown below.
Solution:
Regular polyhedrons include:
 Prisms
 Pyramids
 Platonic solids
The given examples of polyhedron must come under these categories.
Thus, the Egyptian pyramids and Rubik's cube are polyhedrons.
\(\therefore\) Egyptian Pyramids and Rubik's Cube 
Example 5 
Give a few examples of reallife objects in the shape of TORUS.
Solution:
 Donut
 Car Tyres
 Ring
 Polo (mint freshener candy)
 Swimming tube and so on...
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Practice Questions on 3D Shapes
Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.
Maths Olympiad Sample Papers
IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.
You can download the FREE gradewise sample papers from below:
 IMO Sample Paper Class 1
 IMO Sample Paper Class 2
 IMO Sample Paper Class 3
 IMO Sample Paper Class 4
 IMO Sample Paper Class 5
 IMO Sample Paper Class 6
 IMO Sample Paper Class 7
 IMO Sample Paper Class 8
 IMO Sample Paper Class 9
 IMO Sample Paper Class 10
To know more about the Maths Olympiad you can click here
Frequently Asked Questions (FAQs)
1.What is a 3D shape?
In geometry, a threedimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, breadth and height.
2.What is an example of a 3D shape?
Some of the common examples of 3D shapes are:
Dice (cube)
Shoebox (cuboid or rectangular prism)
Ice cream cone (cone)
Globe (sphere)
3.What is the difference between a 2D and a 3D shape?
Twodimensional objects are flat while threedimensional objects have depth in them.
4.How do you teach 3D shapes to kids?
 To teach 3D shapes to kids, we can use various activities to make them understand the three dimensions.
 We can use paper nets to show them the faces, edges and vertices of the solids.