# 3D shapes

3D shapes are nothing but solids that consist of 3 dimensions, namely - length, breadth, and height. The "D" in "3D shapes" stands for "Dimensional."

These 3D shapes occupy space and are found in our day-to-day life. We touch, feel, and use them.

Let us go through this short lesson to know more about 3D shapes!

## Lesson Plan

 1 What Are 3-D Shapes? 2 Tips and Tricks 3 Important Notes on 3D Shapes 4 Solved Examples of 3D Shapes 5 Interactive Questions on 3D Shapes

## What Are 3D Shapes?

In mathematics, we study about 3-dimensional 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; some are in the shape of pyramids or prisms.

Definition of 3-d Shapes

Three-dimensional objects having three dimensions namely length, breadth, and height.

Let us first learn about the 3-dimensional shapes with curved surfaces with examples.

### Sphere

• It is shaped like a ball and is perfectly symmetrical.
• Every point on the sphere is at an equal distance from the center.
• 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 three-dimensional 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 center of the base, a right cone or an oblique cone is formed.

### Torus

• A torus is a regular ring, shaped like a tire or doughnut.
• 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 center of the base, they can be classified into regular and oblique pyramids.
• A pyramid with a:
1. Triangular base is called a Tetrahedron
2. Quadrilateral base is called a square pyramid
3. Pentagon base is called a pentagonal pyramid
4. Regular hexagon base is called a hexagonal pyramid

Let's learn about the three-dimensional shapes called prisms.

### Prisms

• Prisms are solids with identical polygon ends and flat parallelogram sides.
• It has the same cross-section 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 3-dimensional shapes with regular polyhedrons (Platonic Solids).

### Polyhedrons / Platonic solids

• There are five polyhedrons.
1. Tetrahedron with four equilateral-triangular faces
2. Octahedron with eight equilateral-triangular faces
3. Dodecahedron with twelve pentagon faces
4. Icosahedron with twenty equilateral-triangular faces
5. Cube with six square faces
• They have identical faces of regular polygons.

Tips and Tricks
1. Rhyme 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!"

2. Moving your fingers along geometric solids will help you understand the concept of faces, edges, and vertices.

## Properties of 3D Shapes

### Sphere:

Properties Surface Area Volume
• It has no edges or vertices (corners).
• It has one surface.
• It is perfectly symmetrical.
• All points on the surface are the same distance (r) from the center.
$$4\pi r^2$$ $$\frac{4}{3}\pi r^3$$

### Cylinder:

Properties Surface Area  Volume
• It has a flat base and a flat top.
• The bases are always congruent and parallel.
• It has one curved side.
$$2\pi r(r+h)$$ $$\pi r^2h$$

### Cone:

Properties Surface Area Volume
• It has a flat base.
• It has one curved side.
$$\pi r(r+s)$$ $$\frac{1}{3} \pi r^2h$$

### Cube:

Where, edge length (EL)

Properties Surface Area Volume
• It has 6 faces, each with 4 edges (and is a square).
• It has 12 edges.
• It has 8 vertices (corner points) where 3 edges meet.
$$6 \times (E L)^2$$ $$(EL)^3$$

### Prism:

Where, base area (BA), perimeter (P), height (H)

Properties Surface Area Volume
• It has identical ends (polygonal) and flat faces.
• It has the same cross-section all along its length.
$$2\times (BA)$$+ $$P \times H$$ $$BA\times H$$

### Pyramid:

Where base area (BA), perimeter (P) altitude (A), and slant height (SH)

Properties $$Surface$$ $$Area$$ Volume
• Pyramid is a polyhedron with a polygon base and an apex with straight lines.
• Based on its apex alignment with the center of the base, they can be classified into regular and oblique pyramids.

$$B\:A$$ + $$\frac{1}{2} \times P \times(SH)$$

$$\frac{1}{3} \times BA \times A$$

## 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 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

## How to Make 3D Figures

We can better understand the three-dimensional shapes and their properties by using nets.

A 2-dimensional shape that can be folded to form a 3-dimensional 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 three-dimensional 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.

Important Notes
1. Three-dimensional objects have 3 dimensions namely length, breadth, and height.
2. 3D shapes have faces, edges, and vertices.
3. Learning about 3D solids will help us in our day-to-day life as most of our activities revolve and depend on them.

## Solved Examples of 3D Shapes

 Example 1

A construction worker wants to build a 3D sphere using cement.

He wants to know the amount of cement required to construct the sphere of radius 10 inches.

Find the volume of the sphere using the given radius.

Solution

Given,

The radius of the sphere (r) = 10 inches

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:

$$v=\frac{4}{3}\pi\ r^3$$

$$v=\frac{4}{3}\pi (10)^3$$

$$v= 4188.8 \:inches^3$$

 $$\therefore$$ Volume of the cement sphere is $$4188.8\: inches^3$$
 Example 2

Find the surface area of a cuboid of length 3 inches, breadth 4 inches, and height 5 inches.

Solution

Given that,

Length of the cuboid = 3 inches

Breadth of the cuboid = 4 inches

Height of the cuboid = 5 inches

Surface area of the cuboid is

 \begin{align}2 \times (lb + bh + lh) \end{align}

$$= 2 \times (lb + bh + lh)$$

$$= 2(3\times 4 + 4 \times 5+ 3\times 5$$)

$$= 2(12+20+15$$)

$$=2(47)$$

$$=94 \: inches ^2$$

 $$\therefore$$ Surface Area of cuboid = $$94\: inches^2$$
 Example 3

Megan wants to drink milk in a glass which is in the shape of a cylinder.

The height of the glass is 15 inches and the radius of the base is 3 inches.

What is the quantity (volume) of milk that she requires to fill the glass?

Solution

Given that,

Height of the glass  = 15 inches

Radius of the glass base = 3 inches

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 $$in^3$$ of milk to fill her glass.

 $$Volume =424.11\: in^3$$
 Example 4

Identify the regular polyhedron from the images shown below.

Solution

Regular polyhedrons include:

1. Prisms
2. Pyramids
3. 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 real-life objects in the shape of TORUS.

Solution

1. Doughnut
2. Car tires
3. Ring
4. Swimming tube and so on.

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of Is 51 a Prime number. The math journey around Is 51 a Prime number starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

## 1. What can a 3D shape also be called?

In geometry, a three-dimensional shape can also be called a solid.

## 2. What objects have 3D shapes?

The objects that are three-dimensional with length, breadth and height defined are known as 3D Shapes.

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 volume of a 3D shape?

Volume of 3D Shapes refers to the amount of cubic space filled within the shapes.

To find the volume, we need the measurements of the three dimensions.

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