# Solid Shapes

Solid shapes are nothing but solids that consist of 3 dimensions, namely length, breadth, and height. Solid shapes are also known as 3D shapes.

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

In this fun lesson, you can check out some interactive simulations to know more and try your hand at solving a few interesting practice questions at the end of the page. So, what are you waiting for?

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

## Lesson Plan

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

## What Are Solid Shapes?

In mathematics, we study shapes and their different types and try to apply them in real life.

We will now learn about each solid shape in detail.

Solid shapes are classified into several categories.

Some of them have curved surfaces; some are in the shape of pyramids or prisms.

### Definition of Solid Shapes

Solid shapes are three-dimensional shapes that have length, breadth, and height as the three dimensions.

Let us first learn about solid shapes with curved surfaces with examples.

## Different Solid Shapes With Their Properties

### Sphere

Properties Surface Area Volume
• It has no edges or vertices (corners).
• It has one surface.
• It is shaped like a ball and 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.
• It has one curved side.
• The bases are always congruent and parallel.
• It is a three-dimensional object with two identical ends that are either circular or oval.
$$2\pi r(r+h)$$ $$\pi r^2h$$

### Cone

• Based on how the apex is aligned to the center of the base, a right cone or an oblique cone is formed.

Properties Surface Area Volume
• It has a circular or oval base with an apex (vertex).
• It has one curved side.
• A cone is a rotated triangle.
$$\pi r(r+s)$$ $$\frac{1}{3} \pi r^2h$$

### Pyramid

• A pyramid with a:
1. Triangular base is called a Tetrahedron
2. A quadrilateral base is called a square pyramid
3. Pentagon base is called a pentagonal pyramid
4. Regular hexagon base is called a hexagonal pyramid

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

Properties Surface Area Volume
• 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.

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

$$\frac{1}{3} BA^2$$

### Prisms

• 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.

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.
$$2\times (BA)$$+ $$P \times H$$ $$BA\times H$$

### Polyhedrons/Platonic Solids

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. Hexahedron or cube with six square faces.
• They have identical faces of regular polygons.

Where, edge length (EL)

Properties of Cube 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.
$$6 \times (E L)^2$$ $$(EL)^3$$

Tips and Tricks
1. Rhyme to remember solid shapes:

"Solid 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.

## Faces, Edges, and Vertices of Solid Shapes

As mentioned before, solid 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 Solid Shapes

• A face refers to any single flat surface of a solid object.
• Solid shapes can have more than one face.

### Edges of Solid 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 Solid 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:

Solid Shapes Faces Edges Vertices

Sphere

1

0 0
Cylinder

2 0
Cone

1

1 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

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

## Solved Examples

 Example 1

A construction worker wants to build a solid sphere using cement.

He wants to know the amount of cement required to construct a sphere of radius of 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

Peter 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 he 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, he needs roughly 425 $$in^3$$ of milk to fill his glass.

 $$\therefore$$ $$Volume =424.11\: in^3$$
 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 cubes are polyhedrons.

 $$\therefore$$ Egyptian Pyramids and Rubik's Cube

## 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

We hope you enjoyed learning about solid shapes with the simulations and interactive questions. Now you will be able to easily solve problems on different types of solid shapes.

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 is a solid shape?

The objects that are three-dimensional with length, breadth, and height defined are known as solid shapes.

### 2. What can a solid shape also be called?

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

### 3. What are the different solid shapes?

The different solid shapes are cube, cuboid, sphere, prism, cylinder, and pyramid, etc.

### 4. What is the volume of a solid shape?

The volume of solid shapes refers to the amount of cubic space filled within the shapes.

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

### 5. Isasphere solid or flat shape?

A sphere is a solid shape with no edges or vertices (corners).

### 6. Whatis a solid triangle called?

A solid triangle is called a triangular prism.

### 7. Howmany faces does a cube have?

A cube has 6 faces.

### 8.Howmany edges does a cone have?

A cone has 2 edges.

### 9. How many vertices does a cuboid have?

A cuboid has 8 vertices.

### 10. Howmany faces does a tetrahedron have?

A tetrahedron has four equilateral-triangular faces.

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