# NCERT Solutions for Class 7 Mathematics Chapter 15 Visualising Solid Shapes

Class 7 is one of those classes which has great importance in forming a strong base for higher studies of any subject. There are many topics that students of Class 7 learn in Mathematics for the first time.

To help Class 7 students with their Mathematics preparation, Extramarks offers NCERT Solutions for Class 7 Mathematics Chapter 15 where students will find detailed solutions to the questions given in this chapter of their NCERT textbook. Many students find it difficult to solve all the questions in the textbook on their own. These resources will help guide them in the right direction so that they can feel confident when they appear for their final exams.

## NCERT Solutions for Class 7 Mathematics Chapter 15

### NCERT Solutions for Class 7 Mathematics Chapter 15

Chapter 15 of Class 7 NCERT Mathematics textbook introduces students to the different ways of visualising 3D shapes. The chapter includes a lot of interactive exercises that students can try on their own to get a perspective on the topic.. The chapter starts by explaining the differences between 2D and 3D shapes and the relationships between the two. It talks about ‘nets’ which are skeleton- outlines in 2D that can be folded into 3D solid shapes.

In this chapter, students will also learn about the different ways of visualising solid shapes including the shadow view, front-view, side-views, and the top-view.

### Class 7 Chapter 15 Includes

The topics included in Chapter 15 are the following:

• Introduction: Plane figures and Solid shapes
• Faces, Edges, and Vertices
• Nets for Building 3D shapes
• Drawing Solids on a Flat Surface
• Visualising Solid Objects
• Viewing Different Sections of a Solid

The chapter also includes the following exercises. Students can find the solutions to these exercises in NCERT Solutions for Class 7 Mathematics Chapter 15 by Extramarks

 Chapter 15 – Visualizing Solid Shapes Exercises Exercise 15.1 Questions & Solutions Exercise 15.2 Questions & Solutions Exercise 15.3 Questions & Solutions Exercise 15.4 Questions & Solutions

### Introduction

Solid Shapes or figures are very prominent in our surroundings. We come across these solid shapes in the form of mobile phones, laptops, computers, tin cans, ice-cream cones, and many other things. These solid shapes have 3 dimensions ;length, breadth, and height.

### Facts

• Figures drawn on paper are called plane figures, such as circles, triangles, squares, cubes, rectangles, etc.
• The figures that acquire space are called solid shapes, such as spheres, cones, cylinders, cuboids, etc.
• Plane figures are 2-dimensional, and solid shapes are 3-dimensional.
• The corners of a solid are called its vertices, the line segments joining its vertices are called its edges, and its surfaces are called faces.
• 3-D solids can be represented in 2-D by drawing their oblique sketches or isometric sketches.
• A net is a 2-D skeleton outline that results in a 3-D shape when folded. A solid can have more than one net.
• Different sections of a solid are viewed either by slicing it or examining its 2-D shadow. It can also be viewed from the front, top, or side.

### Properties of 2 – Dimensional Shapes

• A 2-dimensional solid has two dimensions, length and breadth.
• The shape of a 2-D solid will always depend on two coordinates.

### Properties of 3 – Dimensional Shapes

• A 3-D shape has length, breadth, and height.
• All things that we see and touch in our surroundings are 3-D solids.
• A surface separates the inner and outside of a 3-D solid.
• 3D solids have faces, vertices, edges, and volume. This property helps you to differentiate between 2-D and 3-D solids.
• Some examples of 3-D shapes are pyramids, cones, spheres, cylinders, prisms, etc.

### Faces, Edges, and Vertices

Face: The flat surface of a solid is called a face.

Edge: A line that joins two corners of a solid is an edge.

Vertices: The corners of a solid are its vertices.

#### Description of Some Basic Shapes

• Square: It has four sides and four corners, and all the sides of a square are of the same length. For example, a napkin, a sandwich, a chessboard, etc.
• Rectangle: A rectangle has four sides and four corners. The opposite sides of a rectangle are of the same length. For example, a tablet, mobile phone, laptop, etc.
• Triangle: A triangle has three sides and three vertices. For example, traffic lights.
• Cuboid: A cuboid has six flat surfaces, twelve straight edges, and eight vertices. For example, books, cabinets, lunch boxes, etc.
• Cube: A cube has six flat faces, eight vertices, and 12 straight edges. For example, sugar cubes, dice, etc.
• Cylinder: A cylinder has three faces, 1 curved face and 2 flat faces. It has two curved edges. For example, a gas cylinder, pipes, candle, etc.
• Cone: A cone has two faces, one slant face and one flat face. For instance, ice-cream cones, funnels, etc.

#### Solid Shape

A solid can be sketched in two ways.

• An oblique sketch is drawn on squared paper and does not reflect exact measurement but expresses all important aspects of the formation of the solid.
• An isometric sketch is drawn on a 3-D drawing paper and has proportional measurements of the solid.

#### Description of Some More Solid Shapes

• Triangular Prism: It resembles a kaleidoscope. It has a triangular base with 5 faces, 9 edges, and 6 vertices.
• Triangular Pyramid: It is also named as a tetrahedron. It has a triangular base with 4 faces, 6 edges, and 4 vertices.
• Square Pyramid: It has a square base with 5 faces, 8 edges, and 5 vertices.
• Sphere: It has no flat face. It has only a spherical face. A sphere has 1 face, no edges, and no vertices.

Polyhedrons

A solid composed of polygon regions is named a polyhedron. For instance, cubes, prisms, cuboids, and pyramids are polyhedrons. Spheres, cylinders, and cones are not polyhedrons as they are not made up of polygon regions. There are two types of polyhedrons- Convex polyhedrons and Regular polyhedrons.

Convex Polyhedrons

When a line segment joining any two points within the surface of a polyhedron lies completely inside or on the shape, the polyhedron is called a convex polyhedron.

Regular Polyhedrons

Regular polygons having identical regular faces which meet at each vertex.

Prism

It is a polyhedron shape whose base and top are congruent, and the lateral faces are parallelograms.

Types of Prism

• Triangular prism
• Rectangular prism
• Pentagonal prism
• Hexagonal prism
• Pyramid
• Triangular pyramid
• Rectangular pyramid
• Square pyramid
• Pentagonal pyramid
• Hexagonal pyramid

Nets for Building a 3 – D Shape

Geometry net is a skeleton outline of a 2-D solid, which results in a 3-D shape when folded. A net can be utilised to find the surface area of an object.

Net is a 2-D representation of a 3-D object unfolded along its edges. A 3-dimensional shape may have many different nets.

#### NCERT Solutions for Class 7 Mathematics

CBSE Class 7 Mathematics consists of 15 chapters. All these chapters are very essential for the students at this level. In addition to the NCERT Solutions for Class 7 Mathematics Chapter 15, you can access solutions for all the other 15 chapters from the Extramarks website and refer to them for daily practise. Here are some of the benefits of referring to NCERT Solutions for Class 7.

• The solutions are prepared by subject experts who have years of experience in teaching.
• All the answers are stated stepwise for quick retention.
• Every answer of every chapter in NCERT Solutions for Class 10 Science is written as per the CBSE guidelines.
• As the explanations are comprehensive, the fundamentals of the chapter are understood by the students in a better way.
• The answers in NCERT solutions are explained in detail, which gives students an idea of how to attempt a question in the board exam in the right manner.

#### NCERT Solutions for Class 7

Along with Mathematics, Extramarks provides NCERT solutions for all the subjects. Students can access the chapter-wise solutions for each subject.

 NCERT Solutions Class 7 Maths Chapter-wise List Chapter 1 – Integers Chapter 2 – Fractions and Decimals Chapter 3 – Data Handling Chapter 4 – Simple Equations Chapter 5 – Lines and Angles Chapter 6 – The Triangle and Its Properties Chapter 7 – Congruence of Triangles Chapter 8 – Comparing Quantities Chapter 9 – Rational Numbers Chapter 10 – Practical Geometry Chapter 11 – Perimeter and Area Chapter 12 – Algebraic Expressions Chapter 13 – Exponents and Powers Chapter 14 – Symmetry Chapter 15 – Visualising Solid Shapes

Ans-

$\left(i\right)\mathrm{The}\mathrm{given}\mathrm{net}\mathrm{can}\mathrm{be}\mathrm{folded}\mathrm{as}:$

$\begin{array}{l}\text{When the faces that are in sky blue colur and in pink colour}\\ \text{are folded to make cube, they will be overlaping each other}\text{.}\end{array}$ $\text{}$ $\text{(ii) The given net can be folded as:}$ $\text{}$

$\text{Thus, a cube can thus be formed in above way}\text{.}$ $\text{}$ $\text{(iii) The given net can be folded as:}$ $\text{}$

$\text{Thus, a cube can thus be formed in above way}\text{.}$ $\text{}$ $\text{(iv) The given net can be folded as:}$ $\text{}$

$\text{Thus, a cube can thus be formed in above way}\text{.}$ $\text{}$ $\text{(v) The given net can be folded as:}$ $\begin{array}{l}\text{}\end{array}$

$\begin{array}{l}\text{When the faces are in blue colour and in red colour are folded}\\ \text{to make a cube, they will be overlapping each other}\text{.}\end{array}$ $\begin{array}{l}\text{(vi) The given net can be folded as:}\end{array}$ $\text{}$

$\text{Thus, a cube can thus be formed in above way}\text{.}$

$\begin{array}{l}\text{Q.2}\end{array}$ $\begin{array}{l}\text{Dice are cubes with dots on each face. Opposite faces of}\\ \text{a die always have a total of seven dots on them.}\end{array}$

$\begin{array}{l}\text{Here are two nets to make dice}\left(\text{cubes}\right)\text{; the numbers}\\ \text{inserted in each square indicate the number of dots in}\\ \text{that box.}\end{array}$

$\begin{array}{l}\text{Insert suitable numbers in the blanks, remembering that}\\ \text{the number on the opposite faces should total to 7.}\end{array}$

Ans-

$\begin{array}{l}\text{(i) The numbers can be inserted as follows so as to make the}\\ \text{given net into a net of a dice}\text{.}\end{array}$

$\begin{array}{l}\text{It can be seen that the sum of opposite faces is 7}\text{.}\\ \\ \text{(ii) The numbers can be inserted as follows so as to make the}\\ \text{given net into a net of a dice}\text{.}\end{array}$ $\text{}$

$\text{It can be seen that the sum of opposite faces is 7}\text{.}$

$\begin{array}{l}\text{Q.3}\end{array}$ $\text{Can this be a net for a die? Explain your answer.}$

Ans-

$\text{The given net can be folded as:}$

$\begin{array}{l}\text{It can be observed that the opposite face of the dice so formed}\\ \text{have 2 and 5, 1 and 4, 3 and 6 on them}\text{.}\\ \text{The sum of numbers on the opposite faces comes to 7, 5 and 9}\\ \text{respectively}\text{.}\\ \text{However, in case of a dice, the sum of numbers on the opposite}\\ \text{faces should be7}\text{.}\\ \text{Therefore, this net is not of a dice}\text{.}\end{array}$

$\begin{array}{l}\text{Q.4}\end{array}$

$\begin{array}{l}\text{Here is an incomplete net for making a cube. Complete}\\ \text{it in at least two different ways. Remember that a cube}\\ \text{has six faces. How many are there in the net here?}\\ \text{(Give two separate diagrams. If you like, you may use a}\\ \text{squared sheet for easy manipulation.)}\end{array}$

Ans-

$\text{There are 3 faces given in the net}\text{. The given net can be completed as:}$

$\begin{array}{l}\text{Q.5}\end{array}$

Match the nets with appropriate solids:

Ans-

$\text{Consider net (i) It can be folded as:}$

$\text{Consider net (ii) It can be folded as:}$

$\text{It is a net of cube}\text{. Hence, (a) is the correct option}\text{.}$ $\text{}$ $\text{Consider net (iii) It can be folded as:}$ $\text{}$

$\text{It is a net of cylinder}\text{. Hence, (b) is the correct option}\text{.}$ $\text{}$ $\text{Consider net (iv) It can be folded as:}$

$\text{It is a net of cone}\text{. Hence, (c) is the correct option}\text{.}$

$\begin{array}{l}\text{Q.6}\end{array}$

$\mathrm{Use}\mathrm{isometric}\mathrm{dot}\mathrm{paper}\mathrm{and}\mathrm{make}\mathrm{an}\mathrm{isometric}\mathrm{sketch}\mathrm{for}\mathrm{each}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{shapes}:$

Ans-

$\begin{array}{l}\text{Q.7}\end{array}$

The dimensions of a cuboid are 5 cm, 3 cm and 2 cm. Draw three different isometric sketches of this cuboid.

Ans-

$\begin{array}{l}\text{Q.8}\end{array}$

Three cubes each with 2 cm edge are placed side by sideto form a cuboid. Sketch an oblique or isometric sketchof this cuboid.

Ans-

$\begin{array}{l}\text{Q.9}\end{array}$

Make an oblique sketch for each one of the given isometric shapes:

Ans-

$\begin{array}{l}\text{Q.10}\end{array}$

Give i an oblique sketch and ii an isometric sketch foreach of the following:a A cuboid of dimensions 5 cm, 3 cm and 2 cm.Is your sketch unique?b A cube with an edge 4 cm long.An isometric sheet is attached at the end of the book.You could try to make on it somecubes or cuboids ofdimensions specified by your friend.

Ans-

$\text{(i) oblique sketch:}$

$\text{(ii) Isometric sketch:}$

$\begin{array}{l}\text{Q.11}\end{array}$

What cross-sections do you get when you give ai vertical cut ii horizontal cut to the following solids?a A brick b Around applec A die d A circular pipee A nice cream cone

Ans-

$\begin{array}{l}\text{(a) A brick}\\ \text{We can give a vertical cut to a brick in the following way:}\end{array}$

$\begin{array}{l}\text{(b) A round apple}\\ \text{We can give a vertical cut to a roudn apple in the following way:}\end{array}$

We can give a horizontal cut to a round apple in the following way:

(c) A die

We can give a vertical cut to a die in the following ways:

$\text{We can give a horizontal cut to a die in the following way:}$ $\begin{array}{l}\text{}\end{array}$

$\begin{array}{l}\text{(d) A circular pipe}\\ \text{We can give a vertical cut to a circular pipe in the following way:}\end{array}$

$\text{We can give a horizontal cut to a circular pipe in the following way:}$ $\begin{array}{l}\text{}\end{array}$

$\begin{array}{l}\text{(e) An ice cream cone}\\ \text{We can give a vertical cut to an ice cream cone in the following way:}\end{array}$

$\mathrm{We}\mathrm{can}\mathrm{give}\mathrm{a}\mathrm{horizontal}\mathrm{cut}\mathrm{to}\mathrm{an}\mathrm{ice}\mathrm{cream}\mathrm{cone}\mathrm{in}\mathrm{the}\mathrm{following}\mathrm{way}:$

$\begin{array}{l}\text{Q.12}\end{array}$

A bulb is kept burning just right above the following solids.Name the shape of the shadows obtained in each case.Attempt to give a rough sketch of the shadow.(You maytry to experiment first and then answer these questions).

Ans-

$\begin{array}{l}\text{The shapes of the shadows of these figures will be as follows:}\\ \text{(i) A ball}\end{array}$

$\text{The shape of the shadow of a ball will be a circle}\text{.}$ $\text{(ii) A cylindrical pipe}$

$\text{The shape of the shadow of a cylindrical pipe will be a rectangle}$ $\text{}\text{.}$ $\text{}$ $\text{(iii) A book}$

$\text{}$ $\text{The shape of the shadow of a book will be a rectangle}\text{.}$

$\begin{array}{l}\text{Q.13}\end{array}$

$\begin{array}{l}\text{Here are the shadows of some 3-D objects, when seen}\\ \text{under the lamp of an over head projector. Identify the}\\ \text{solid}\left(\text{s}\right)\text{that match each shadow. (There may be}\\ \text{multipleanswers for these!).}\end{array}$

Ans-

$\begin{array}{l}\text{The given shadows can be obtained in case of the following}\\ \text{objects}\text{.}\\ \text{(i) Compact disk (Sphere)}\\ \text{(ii) A}\text{​}\text{dice (Cube)}\\ \text{(iii) Triangular pyramid (Cone)}\\ \text{(iv) Note Book (Cuboid)}\end{array}$

$\begin{array}{l}\text{Q.14}\end{array}$

$\begin{array}{l}\text{Examine if the following are true statements:}\\ \left(\text{i}\right)\text{The cube can cast a shadow in the shape of a rectangle.}\\ \left(\text{ii}\right)\text{The cube can cast a shadow in the shape of a hexagon.}\end{array}$

Ans-

$\begin{array}{l}\text{A cube can cast shadow only in the shape of a square}\text{.}\\ \text{Therefore, any other shapes are not possible}\text{.}\\ \text{So,}\\ \text{(i) False}\\ \text{(ii) False}\end{array}$