Exercise 13.2

1.Count the number of faces , vertices , and edges of given polyhedra and verify Euler’s formula.

Solution

Triangular Prism Analysis

Faces (F):

triangular faces

rectangular faces

Vertices (V):

vertices on the top triangle

vertices on the bottom triangle

Edges (E):

edges on the top triangle

edges on the bottom triangle

edges connecting the top and bottom

Euler's Formula Verification

Euler's formula: F - E + V =

Plugging in the values:

- + = 2

= 2

Therfore Euler's formula is verified for the triangular prism.

ii.

Solution

Polyhedron Analysis

Faces (F):

top triangular face

bottom pentagonal face

rectangular faces

Vertices (V):

vertices on the top triangle

vertices on the bottom pentagon

Edges (E):

edges on the top triangle

edges on the bottom pentagon

edges connecting the top and bottom

Euler's Formula Verification

Euler's formula: F - E + V = 2

Plugging in the values:

- + =

= 2

Therefore Euler's formula is verified for the polyhedron.

iii.

Solution

Hexagonal Prism Analysis

Faces (F):

hexagonal faces

rectangular faces

Vertices (V):

vertices on the top hexagon

vertices on the bottom hexagon

Edges (E):

edges on the top hexagon

edges on the bottom hexagon

edges connecting the top and bottom

Euler's Formula Verification

Euler's formula: F - E + V = 2

Plugging in the values:

- + = 2

= 2

Therefore Euler's formula is verified for the hexagonal prism.

iv.

Solution

Polyhedron Analysis

Faces (F):

1 pentagonal face

triangular faces

Vertices (V):

vertices on the pentagon

vertex at the bottom

Edges (E):

edges on the pentagon

edges connecting to the bottom

Euler's Formula Verification

Euler's formula: F - E + V = 2

Plugging in the values:

- + = 2

= 2

Therefore Euler's formula is verified for the polyhedron.

Solution

Polyhedron Analysis

Faces (F):

square face

triangular faces

Vertices (V):

vertices on the square

vertex at the top

Edges (E):

edges on the square

edges connecting to the top

Euler's Formula Verification

Euler's formula: F - E + V = 2

Plugging in the values:

- + = 2

= 2

Therefore Euler's formula is verified for the polyhedron.

vi.

Solution

Polyhedron Analysis

Faces (F): (all hexagons)

Vertices (V):

Edges (E):

Euler's Formula Verification

Euler's formula: F - E + V = 2

Plugging in the values:

- + = 2

= 2

Therefore Euler's formula is verified for the polyhedron.

vii.

Solution

Polyhedron Analysis

Faces (F): (all triangles)

Vertices (V):

Edges (E):

Euler's Formula Verification

Euler's formula: F - E + V = 2

Plugging in the values:

- + = 2

= 2

Therefore Euler's formula is verified for the polyhedron.

viii.

Solution

Polyhedron Analysis

Faces (F):

square faces

triangular face

Vertices (V):

Edges (E):

Euler's Formula Verification

Euler's formula: F - E + V = 2

Plugging in the values:

- + = 2

= 2

Therefore Euler's formula is verified for the polyhedron.

2. Is a square prism and cube are same? explain.

Solution

Square Prism Image:

Cube Image:

Is a square prism and cube are same.

Here's why:

Square Prism:

Faces: (2 square bases, 4 rectangular sides)

Edges:

Vertices:

Shape of Bases:

Shape of Sides:

Cube:

Faces:

Edges:

Vertices:

Shape of Bases:

Shape of Sides:

The main difference lies in the shape of their sides. A square prism has rectangular sides, while a cube has square sides. This means that all edges of a cube are the same length, whereas a square prism can have different lengths for its edges.

3. Can a polyhedra have 3 triangular faces only? explain.

Solution

Can a polyhedra have 3 triangular faces only?

A polyhedron needs at least three faces to meet at each corner to be a 3D shape. If you only have 3 triangles, they can only make a flat triangle or a floppy, open shape, not a closed 3D one. You'd need more faces to make a proper corner.

4. Can a polyhedra have 4 triangular faces only? explain.

Solution

Can a polyhedra have 4 triangular faces only?

Yes! This shape is called a triangular pyramid (or tetrahedron). It has a triangular base and triangular sides that meet at a point.

5. Complete the table by using Euler’s formula.

Faces (F)	Vertices (V)	Edges (E)
8	6	?
5	?	9
?	12	30

Solution

Let's break down how we found the missing values:

Row 1: We had F = and V = .

Using E = F + V - 2, we get E = + - 2 = .

Row 2: We had F = and E = .

Using V = 2 + E - F, we get V = 2 + - = .

Row 3: We had V = and E = .

Using F = 2 + E - V, we get F = 2 + - = .

Threfore,

Faces (F)	Vertices (V)	Edges (E)
8	6
5		9
	12	30

6. Can a polyhedra have 10 faces, 20 edges and 15 vertices?

Solution

Can a polyhedra have 10 faces, 20 edges and 15 vertices?

Euler's formula for polyhedra states: F - E + V = 2

Let's plug in the given values: - + =

Since the result is not , this combination of faces, edges, and vertices does not satisfy Euler's formula, and therefore cannot form a polyhedron.

7. Complete the following table:

Object	No. of vertices	No. of edges

Solution

Explanation:

To find the number of vertices and edges, we carefully examined each 3D shape:

Rectangular Prism:

Vertices (V): Count the corners: vertices

Edges (E): Count the lines: edges

Square Pyramid:

Vertices (V): Count the corners: vertices

Edges (E): Count the lines: edges

Triangular Prism:

Vertices (V): Count the corners: vertices

Edges (E): Count the lines: edges

Object	No. of vertices	No. of edges
	8	12
	5	8
	6	9

8. Name the 3-D objects or shapes that can be formed from the following nets .

(i).

Solution

Pyramid

(ii).

Solution

(iii).

Solution

(vi).

Solution

(v).

Solution

(vi).

Solution

(vii).

Solution

9 (i). Draw the following diagram on the check ruled book and find out which of the following diagrams makes cube ?

(a).

Solution

It can form a cube.

Cannot form a cube: The squares are arranged in a straight line and won't create a closed shape when folded.

(b).

Solution

It can form a cube.

Can form a cube: This is a classic net that forms a cube.

(c).

Solution

It can form a cube.

Cannot form a cube: When folded, there will be overlapping faces and a missing face.

(d).

Solution

It can form a cube.

Cannot form a cube: There will be overlapping faces and a missing face.

(e).

Solution

It can form a cube.

Can form a cube: This is another standard net that forms a cube.

(f).

Solution

It can form a cube.

Can form a cube: This is also a valid net for a cube.

(g).

Solution

It can form a cube.

Cannot form a cube: When folded, there will be overlapping faces and a missing face.

(h).

Solution

It can form a cube.

Cannot form a cube: When folded, there will be overlapping faces and a missing face.

(i).

Solution

It can form a cube.

Cannot form a cube: When folded, there will be overlapping faces and a missing face.

(j).

Solution

It can form a cube.

Cannot form a cube: When folded, there will be overlapping faces and a missing face.

(k).

Solution

It can form a cube.

Cannot form a cube: When folded, there will be overlapping faces and a missing face.

(ii). Answer the following questions.

(a) Name the polyhedron which has four vertices, four faces?

Solution

(also known as a triangular pyramid)

(b) Name the solid object which has no vertex?

Solution

(c) Name the polyhedron which has 12 edges?

Solution

(d) Name the solid object which has one surface?

Solution

(e) How a cube is different from cuboid?

Solution

A cube has all its sides as , while a cuboid can have sides.

(f) Name the two shapes which have the same number of edges, vertices and faces.

Solution

(g) Name the polyhedron which has 5 vertices and 5 faces?

Solution

(iii). Write the names of the objects given below.

(a)

Solution

(b)

Solution

(c)

Solution

(d)

Solution

Chapter 13: Visualising 3D in 2D

Glossary

Exercise 13.2

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Chapter 13: Visualising 3D in 2D

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Glossary

Exercise 13.2