Visualising Solid Shapes - Class 8 Maths - Extra Questions

Euler's Formula is $$F + V - E = 2$$ ,  

Euler's Formula is $$F + V - E = 2$$ ,  where $$F$$ = number of faces, 

For a right prism the number of lateral surfaces will always be equal to the number of edges of the base. 

For any polyhedron that doesn't intersect itself; 

Number of Faces + the Number of Vertices (corner points) - the Number of Edges is always equal to 2.

This can also be written as:   $$F + V - E = 2$$

Therefore, $$k=2$$

Regular Polyhedrons are the solids with identical regular polygons as their faces.
Example: A cube is a platonic solid because all six of its faces are congruent squares.

A convex polyhedron is one in which all faces make it convex

Euler's formula, $$F + V = E + 2$$
$$\therefore 6 + 8 = E + 2$$
$$14 = E + 2$$
$$\therefore E = 14 - 2$$
$$\therefore E = 12$$

Here $$F=5$$
$$V=5$$
$$E=8$$
$$F+V=5+5=10.....(i)$$
$$E+2=8+2=10.....(ii)$$
From $$(i)$$ and $$(ii)$$
$$F+V=E+2$$

Total length $$(L)$$ of the edges of a cube = Sum of the lengths of all $$(12)$$ edges of the cube.
$$=l+l+l+l+l+l+l+l+l+l+l+l=12l$$

No, such a polyhedron is not possible. A polyhedron has minimum $$4$$ faces.

A rectangular pyramid has $$5$$ faces, $$8$$ edges and $$5$$ vertices.

As per question:
Faces = 3, Vertices = 0, and Edges = 2.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$3 + 0 - 2 = 2$$
          $$3 - 2 = 2$$
          $$1 \neq 2$$
Therefore, these given value do not satisfy Euler's formula. So, the given figure is not a polyhedral.

As per question:
Faces = 6, Vertices = 8, and Edges = 12.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$6 + 8 - 12 = 2$$
          $$14 - 12 = 2$$
          $$2 = 2$$
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.

As per question:
Faces $$= 5$$ , Vertices $$= 6$$ , and Edges $$= 9$$ .
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$5 + 6 - 9 = 2$$
          $$11 - 9 = 2$$
          $$2 = 2$$
Therefore, these given values satisfy Euler's formula. So, the given figure is a polyhedron.

According to the figure,
Faces = 3, Vertices = 0, and Edges = 2.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$3 + 0 - 2 = 2$$
          $$1 \neq 2$$
Therefore, these given value do not satisfy Euler's formula. So, the given figure is not a polyhedral.

 According to the figure,
Faces = 11, Vertices = 11, and Edges = 20.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$11 + 11 - 20 = 2$$
          $$22 - 20 = 2$$
          $$2 = 2$$
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.

 Faces = 5, Vertices = 6, and Edges = 9.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$5 + 6 - 9 = 2$$
          $$11 - 9 = 2$$
          $$2 = 2$$
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.

 According to the figure,
Faces = 9, Vertices = 9, and Edges = 16.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$9 + 9 - 16 = 2$$
          $$18 - 16 = 2$$
          $$2 = 2$$
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.

According to the figure,
Faces = 7, Vertices = 10, and Edges = 15.
By putting the given values in Euler's formula,
          $$F + V - E = 2$$
          $$7+10-15 = 2$$
          $$17-15 = 2$$
          $$2 = 2$$
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.

The shape of the book is generally cuboids and cuboids look-alike rectangular prism. A figure that represents the mathematics textbook is shown above.

We know that,
Euler's Formula is valid for any polyhedrons. i.e.
$$F +V - E =2$$
Given, $$F = 40$$ and $$V = a$$ and $$E = 60$$
According to the formula:
$$40 + a - 60 = 2$$
$$a - 20 = 2$$
$$a = 20 + 2$$
$$a = 22=$$ number of vertices

As per question:
  Euler's Formula is for any polyhedrons. i. e.
   F + V - E = 2
   Where, F = Faces and V = Vertices and E = Edges
i) Given, F = 7 and V = 10 and E = x
  According to the formula:
      7 + 10 - x = 2
      17 - x = 2
      -x = 2 - 17
       x = 15


ii) Given, F = y and V = 12 and E = 18
  According to the formula:
      12 + y - 18 = 2
      -6  + y = 2
      y = 2 + 6
       y = 8

iii) Given, F = 9 and V = z and E = 16
  According to the formula:
      9 + z - 16 = 2
      -7 +  z = 2
      z = 2 + 7
       z = 9  

iv) Given, F = p and V = 6 and E = 12
  According to the formula:
      p + 6 - 12 = 2
      p - 6 = 2
      p = 2 + 6
       p = 8

v) Given, F = 6 and V = q and E = 12
  According to the formula:
      6 + q - 12 = 2
      -6 + q = 2
      q= 2 + 6
       q = 8

vi) Given, F = 8 and V = 11 and E = r
  According to the formula:
      8 + 11 - r = 2
      19 - r = 2
      r = 19 - 2
       r = 17

According to the figure,
Faces = 2, Vertices = 1, and Edges = 0.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$2 + 1 - 0 = 2$$
          $$3 - 0 = 2$$
          $$3 \neq 2$$
Therefore, these given value do not satisfy Euler's formula. So, the given figure is not a polyhedral.

According to the figure,
Faces = 1, Vertices = 0, and Edges = .
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$1 + 0 - 0 = 2$$
          $$1 - 0 = 2$$
          $$1 \neq 2$$
Therefore, these given value do not satisfy Euler's formula. So, the given figure is not a polyhedral.

According to the figure,
Faces = 8, Vertices = 12, and Edges = 18.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$8 + 12 - 18 = 2$$
          $$20 - 18 = 2$$
          $$2 = 2$$
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.

According to the figure,
Faces = 10, Vertices = 16, and Edges = 24.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$10 + 16 - 24 = 2$$
          $$26 - 24 = 2$$
          $$2 = 2$$
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.

According to the figure,
Faces = 8, Vertices = 6, and Edges = 12.
By putting the given values in Euler's formula,
          $$F + V- E = 2$$
          $$8 + 6 - 12 = 2$$
          $$14 - 12 = 2$$
          $$2 = 2$$
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.

As per question:
Euler's Formula is for any polyhedrons. i.e.
F + V - E = 2
Given, F = 9 and V = 9 and E = 16
   According to the formula:
   9 + 9 - 16 = 2
   18 - 16 = 2
    2 = 2
Therefore, these given value satisfy Euler's formula. So, the given figure is a polyhedral.
Now, as per given data the figure shown below:

This shown figure is octagonal pyramid.

Polyhedron	Faces(F)	Vertices(V)	Edges(E)	F-E+V
Square Pyramid	$$5$$	$$5$$	$$8$$	$$2$$

Number of faces = $$5$$

Number of edges = $$8$$

Number of vertices = $$5$$

Hence,

$$F – E + V = 5 – 8 + 5$$

$$= – 3 + 5$$

$$= 2$$

Polyhedron	Faces(F)	Vertices(V)	Edges(E)	F-E+V
Hexagonal Prism	$$8$$	$$12$$	$$18$$	$$2$$

A hexagonal prism has,

Number of faces = $$8$$

Number of edges = $$18$$

Number of vertices = $$12$$

Hence,

$$F – E + V = 8 – 18 + 12$$

$$= – 10 + 12$$

$$= 2$$

Polyhedron	Faces(F)	Vertices(V)	Edges(E)	F-E+V
Hexahedron	$$8$$	$$6$$	$$12$$	$$2$$

Number of faces = $$8$$

Number of edges = $$12$$

Number of vertices = $$6$$

Hence,

$$F – E + V = 8 – 12 + 6$$

$$= – 4 + 6$$

$$= 2$$

A triangular pyramid has,

Number of faces = $$4$$

Number of edges = $$6$$

Number of vertices = $$4$$

Hence,

$$F – E + V = 4 – 6 + 4$$

$$= – 2 + 4$$

Polyhedron	Faces(F)	Vertices(V)	Edges(E)	F-E+V
Triangular Pyramid	$$4$$	$$4$$	$$6$$	$$2$$

$$(i)F+V-E=2$$

$$x+15-20=2$$

$$x-5=2$$
$$x=7$$

$$(ii)F+V-E=2$$

$$6+y-8=2$$

$$y-2=2$$
$$y=4$$

$$(iii)F+V-E=2$$

$$14+26-z=2$$

$$40-z=2$$
$$z=38$$

Euler's Formula is $$F + V - E = 2$$ ,  where $$F$$ = number of faces

$$V$$ = number of vertices, and $$E$$ = number of edges.

So, $$12 + V - 30 = 2$$


$$\implies  V = 30+2-12$$

   Hence       $$V = 20$$

A polyhedron is a solid in $$3-D$$ with flat polygonal faces, straight edges and sharp corners or vehicles.

Suppose w.l.o.g. that the plane $$\alpha$$ through KL meets the interiors of edges AC and BD at X and Y. Let $$\vec {AX}=\lambda \vec {AC}$$ and $$\vec {BY}=\mu \vec {BD}$$ , for $$0\leq \lambda, \mu \leq 1$$ . Then the vector $$\vec {KX}=\lambda \vec {AC}-\frac {\vec {AB}}{2}, \vec {KY}=\mu \vec {BD}+\frac {\vec {AB}}{2}, \vec {KL}=\frac {\vec {AC}}{2}+\frac {\vec {BC}}{2}$$ are coplanar; i.e., there exist real numbers a, b, c, not all zero, such that
$$\vec 0=a\vec {KX}+b\vec {KY}+c\vec {KL}=(\lambda a+\frac {c}{2})\vec {AC}+(\mu b+\frac {c}{2})\vec {BC}+(\frac {b-a}{2})\vec {AB}$$
Since $$\vec {AC}, \vec {BD}, \vec {AB}$$ are linearly independent, we must have $$a=b$$ and $$\lambda =\mu$$ . We need to prove that the volume of the polyhedron KXLYBC, which is one of the parts of the tetrahedron ABCD partitioned by $$\alpha$$ , equals half of the volume V of ABCD. Indeed, we obtain
$$VKXLYBC=VKXLC+VKBYLC=\frac {1}{4}(1-\lambda)V+\frac {1}{4}(1+\mu)V=\frac {1}{2}V$$

1.Here,

No. of Faces (F):5

No. of Vertices (V):6

No. of Edges (E):9

Euler’s formula: F + V = E + 2

⇒ 5 + 6 = 9 + 2

⇒ 11 = 11

∴ Euler’s formula satisfies for the shape.

2.Here,

No. of Faces (F):7

No. of Vertices (V):10

No. of Edges (E):15

Euler’s formula: F + V = E + 2

⇒ 7 + 10 = 15 + 2

⇒ 17 = 17

∴ Euler’s formula satisfies for the shape.

3.Here,

No. of Faces (F):8

No. of Vertices (V):12

No. of Edges (E):18

Euler’s formula: F + V = E + 2

⇒ 8 + 12 = 18 + 2

⇒ 20 = 20

∴ Euler’s formula satisfies for the shape.

4.Here,

No. of Faces (F):6

No. of Vertices (V):6

No. of Edges (E):10

Euler’s formula: F + V = E + 2

⇒ 6 + 6 = 10 + 2

⇒ 12 = 12

∴ Euler’s formula satisfies for the shape.

5.Here,

No. of Faces (F):5

No. of Vertices (V):5

No. of Edges (E):8

Euler’s formula: F + V = E + 2

⇒ 5 + 5 = 8 + 2

⇒ 10 = 10

∴ Euler’s formula satisfies for the shape.

No. of Faces (F):8

No. of Vertices (V):12

No. of Edges (E):18

Euler’s formula: F + V = E + 2

⇒ 8 + 12 = 18 + 2

⇒ 20 = 20

∴ Euler’s formula satisfies for the shape.

7.Here,

No. of Faces (F):8

No. of Vertices (V):6

No. of Edges (E):12

Euler’s formula: F + V = E + 2

⇒ 8 + 6 = 12 + 2

⇒ 14 = 14

∴ Euler’s formula satisfies for the shape.

8.Here,

No. of Faces (F):7

No. of Vertices (V):10

No. of Edges (E):15

Euler’s formula: F + V = E + 2

⇒ 7 + 10 = 15 + 2

⇒ 17 = 17

∴ Euler’s formula satisfies for the shape.

The possible polyhedrons with 12 edges are a cube, cuboid, Hexagonal Pyramid.

In square pyramid, there is square base and on each side of a base there lies a triangle.

By, Euler's formula, we know that
$$V-E+F=2$$
Here, $$E=10$$ and $$F=6$$
$$\Rightarrow V-10+6=2$$

$$L$$ is the midpoint of $$AB$$

A Cuboid may be the square prism.

Consider given the figure,

Now,

Number of face in 1^st figure= $$3+2=5$$

Number of face in 2nd figure= $$5+2=7$$

Number of face in 3^{rd figure}  $$=6+2=8$$

Number of face in 4rt figure $$=5+1=6$$

 

Hence, this is the answer.

Number of vertices $$=5$$

We know that, $$Euler\, \, formula=F+V=E+2$$

The net has $$2$$ triangular faces and $$3$$ rectangular faces.

Faces	$$a$$	$$5$$	$$20$$	$$6$$
Vertices	$$6$$	$$b$$	$$12$$	$$d$$
Edges	$$12$$	$$9$$	$$c$$	$$12$$

 
(i) We can write it as 
$$a + 6 - 12 = 2$$  
By further calculation 
$$a = 2 - 6 + 12$$  

(ii) We can write it as 
$$b + 5 - 9 = 2$$  
By further calculation
$$b = 2 + 9 - 5$$  
$$b = 6$$  

(iii) We can write it as 
$$20 + 12 - c = 2$$  
By further calculation 
$$32 - c = 2$$  
$$c = 32 - 2 = 30$$  

(iv) We can write it as 
$$6 + d - 12 = 2$$  
By further calculation 
$$d - 6 = 2$$  
$$d = 2 + 6 = 8$$

From the question it is given that, a model of a ship is made to a scale of $$1 : 250$$
(i) Given, the length of the model is $$1.6 m$$
Then, length of the ship $$= (1.6 \times 250)/1$$
$$= 400 m$$
(ii) Given, the area of the deck of the ship is $$2.4 m^{2}$$
Then, area of deck of the model $$= 2.4 \times (1/250)^{2}$$
$$= 1,50,000 m^{2} = 4 m^{2}$$
(iii) Given, the volume of the model is $$1 km^{3}$$
Then, Volume of ship $$= (1/2503) \times 1 km^{3}$$
$$= 1/(250)^{3} \times 1000^{3}$$
$$= 4^{3}$$
$$= 64 m^{3}$$
Therefore, volume of ship is $$64 m^{3}$$ .

Given, perimeter of the Pyramid $$=96cm$$
Height $$= 16cm$$

a. Let side be $$a\ cm$$

In a $$3-$$ dimensional figure, let the number of faces be $$F$$ , the number of edges be $$E$$ and the number of vertices be $$V$$ .

Shape	Faces	Vertices	Edges	$$F-E+V$$
Cube	$$6$$	$$8$$	$$12$$	$$2$$
Ocrahedron	$$8$$	$$6$$	$$12$$	$$2$$

A triangular prism has 2 triangular faces, 3 rectangular faces, 9 edges and 6 vertices.

As per question:
Euler's Formula is for any polyhedrons. i.e.
$$F + V - E = 2$$
Given, $$F = 8 ,V = 12$$ and $$E = 6$$
According to the formula:
$$8 + 12 -6 =2$$
$$20 - 6 = 2$$
$$14 \neq 2$$
Therefore, these given values do not satisfy Euler's formula. So, the given figure is not a polyhedral.

Octagonal Pyramid: It has 16 edge. The structure of octagonal pyramid is shown below:

Hexagonal Prism: It has 18 edge. The structure of hexagonal prism is shown below:

The structure of the hexagonal prism is shown above. From the figure, we can say that a hexagonal prism has $$12$$ vertices.

$$\textbf{Step 1: Calculate Number of Edges for a Polyhedron}$$

                  $$\text{Faces(F) = 7 and Vertices(V) }$$ $$= 10$$           $$\textbf{[Given]}$$

                  $$\text{According to Euler's formula,}$$

                  $$F+V=E+2$$   $$\text{(Where F,V and E are Number of Faces ,Vertices and Edges)}$$

                 $$\Rightarrow F + V - E = 2$$

$$\textbf{Therefore,Number of Edges in Polyhedron are 15.}$$ 

$$F+V-E=2$$

$$x+15-20=2$$

$$x-5=2$$
$$⇒x=2+5=7$$

Number of vertices $$= 9$$

Euler’s formula,
$$F+V-E=2$$

$$(i)x+15-20=2$$
$$\Rightarrow x=2+5=7$$

$$(ii)6+y-8=2$$

$$\Rightarrow y=2+2=4$$

$$(iii)14+26-z=2$$
$$\Rightarrow z=14+26-2=38$$

Pentagonal pyramid

Examples of pyramid are as follows: