Visualising Solid Shapes Test

Result

Visualising Solid Shapes Test - 16

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Question 1
1 / -0

A pentagonal pyramid has 6 faces, 6 vertices and 10 edges. Find its Euler characteristic.

A
1

B
2

C
3

D
4

Solution

The Euler's formula for polyhedron is
$$V - E + F =\chi$$
Where
$$\chi$$ = Euler characteristic
$$V$$ = number of vertices = $$6$$
$$E$$ = number of edges = $$10$$
$$F$$ = number of faces = $$6$$
So,
$$6-10+6=\chi$$
$$\chi = 12-10$$
$$\chi = 2$$
Question 2
1 / -0

If you look at this object from the top, what will you see?

A

B

C

D

Solution

We will see the first figure as top view.
Question 3
1 / -0

Using Euler's formula, find the number of vertices for a given figure with 6 faces and 12 edges.

A
8

B
10

C
12

D
2

Solution

Euler's polyhedron formula, $$V - E + F =2$$
$$V$$ = number of vertices = ?
$$E$$ = number of edges = $$12$$
$$F$$= number of faces = $$6$$
$$V-12+6=2$$
$$V = 6+2$$
$$V = 8$$
Number of vertices = $$8$$
Question 4
1 / -0

Apply Euler's formula for a solid that has 16 faces and 21 edges. How many vertices does it have?

A
$$3$$

B
$$5$$

C
$$7$$

D
$$9$$

Solution

We know that, Euler's polyhedron formula is $$V - E + F =2$$
where, $$V$$ is the number of vertices
$$E$$ is the number of edges
$$F$$ is the number of faces

Here, $$E=16,\ F=21$$
Hence, $$V-21+16=2$$
$$\Rightarrow V -5 = 2$$
$$\Rightarrow V = 7$$

Hence, the solid has $$7$$ vertices.
Question 5
1 / -0

What is the value of F + V - E for a hexagonal prism?

A
5

B
1

C
4

D
2

Solution

Euler's polyhedron formula, $$V - E + F =2$$
A hexagonal prism has $$8$$ faces, $$F = 8$$
A hexagonal prism has $$12$$ vertices, $$V = 12$$
A hexagonal prism has $$18$$ edges, $$E = 18$$
So, $$12 -18+8=2 $$
Question 6
1 / -0

A octahedron has _____ faces and 6 vertices with 12 edges. (Use Euler's formula)

A
4

B
8

C
12

D
24

Solution

Euler's polyhedron formula, $$V - E + F =2$$
$$V$$ = number of vertices = $$6$$
$$E$$ = number of edges = $$12$$
$$F$$ = number of faces = ?
$$6-12+F=2$$
$$F = 6+2$$
$$F = 8$$
Number of faces = $$8$$
A octahederon has$$8$$ faces and $$6$$ vertices with $$12$$ edges.
Question 7
1 / -0

Which of the following diagrams represents the front view of the given solid?

A

B

C

D

Solution

The second option represents the front view of the given solid.
Question 8
1 / -0

How many faces does a solid figure has, if their vertices is $$13$$ and edges is $$24$$?

A
$$12$$

B
$$13$$

C
$$14$$

D
$$15$$

Solution

Euler's polyhedron formula, $$V - E + F =2$$
$$V$$ $$=$$ number of vertices $$=$$ $$13$$
$$E$$ $$=$$ number of edges $$=$$ $$24$$
$$F$$ $$=$$ number of faces $$= ?$$
Therefore, $$13-24+F=2$$
$$\Rightarrow F = 11+2$$
$$\Rightarrow F = 13$$
Number of faces $$=$$ $$13$$
Question 9
1 / -0

A square based pyramid has $$5$$ faces and $$5$$ vertices. Find the number of edges.

A
$$6$$

B
$$8$$

C
$$10$$

D
$$12$$

Solution

Euler's polyhedron formula, $$V - E + F =2$$
$$V$$ $$=$$ number of vertices $$=$$ $$5$$
$$E$$ $$=$$ number of edges $$= ?$$
$$F$$ $$=$$ number of faces $$=$$ $$ 5$$
Thus $$5-E+5=2$$
$$\Rightarrow E = 10-2$$
$$\Rightarrow E = 8$$
Number of edges $$=$$ $$8$$
Question 10
1 / -0

Determine whether the given figure is an example of

A
polyhedron

B
polygon

C
triangle

D
pentagon

Solution

Polyhedron is a solid with flat faces.
So, the given figure is an example of polyhedron.