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8. Can a polyhedron have 10 faces, 20 edges and 15 vertices?

Checking using Euler's Formula: F + V = E +2; Given that F = 10, V = 15 and E = 20 so, Putting in equation: 10 + 15 = 35  and 20 +2 = 22 Hence, given polyhedron having 10 faces, 20 edges and 15 vertices is not possible.

7. Using Euler’s formula find the unknown.

    

We know the Euler Formula: F + V = E + 2: hence we can easily find the unknown let say 'F, V, and E' So, For Ist case:  F + 6  = 12 + 2  or  F = 8, hence the number of faces in Ist case is 8. For IInd Case; 5 + V = 9 + 2   or  V = 6, hence the number of vertices in IInd case is 6. For IIIrd case; 20 + 12 = E + 2     or   E = 30, hence the number of Edges in the IIIrd case is...

6. Verify Euler’s formula for these solids.

    

(ii) Euler formula F+V = E + 2. So, Number of faces (F) = 9, Number of vertices (V)  = 9 and number of edges (E)  = 16. Hence putting in the formula; we get, 9 + 9 = 16 + 2 18 = 18 hence, it satisfies the Euler formula.

6. Verify Euler’s formula for these solids.

    

(i) Euler formula F+V = E + 2. So, Number of faces (F) = 7, Number of vertices (V)  = 10 and number of edges (E)  = 15. Hence putting in the formula; we get, 7 + 10 = 15 + 2 17 = 17 hence, it satisfies the Euler formula.  

5. Is a square prism same as a cube? Explain.

No, not always, because it can be a cuboid also when extended.

4. (i) How are prisms and cylinders alike?

(i) A prism is a polyhedron whose base and top are congruent polygons and whose other faces, i.e., lateral faces are parallelograms in shape. Also, the cylinder has base on top and bottom and faces are parallel.

3. Which are prisms among the following?

    

(iv) Given box is a prism.

3. Which are prisms among the following?

    

(iii) Table weight is not a prism.  

3. Which are prisms among the following?

    

(ii) The unsharpened pencil is a prism.

3. Which are prisms among the following?

        

(i) a nail is not a prism.  

2. Is it possible to have a polyhedron with any given number of faces? (Hint: Think of a pyramid).

Yes, it is possible if the number of faces is four or more than four.

1. Can a polyhedron have for its faces

    (iii) a square and four triangles?

(iii) Yes, a polyhedron can have faces of a square and four triangles which makes a pyramid on square base.

1. Can a polyhedron have for its faces

    (ii) 4 triangles?

(ii) Yes, a polyhedron having 4 triangles is pyramid on triangular base.

1. Can a polyhedron have for its faces

    (i) 3 triangles?

(i) No, a polyhedron cannot have 3 triangles for its faces. 

Tabulate the number of faces, edges and vertices for the following polyhedrons: (Here ‘V’ stands for number of vertices, ‘F’ stands for number of faces and ‘E’ stands for number of edges).

Solid F V E F+V E+2
Cuboid          
Triangular pyramid          
Triangular prism          
Pyramid with square base          
Prism with square base          

What do you infer from the last two columns? In each case, do you find
F + V = E + 2, i.e., F + V – E = 2? This relationship is called Euler’s formula.
In fact this formula is true for any polyhedron.

    Solid F V E F+V E+2 Cuboid 6 8 12 14 14 Triangular pyramid 4 4 6 8 8 Triangular prism 5 6 9 11 11 Pyramid with square base 5 5 8 10 10 Prism with square base 6 8 12 14 14

4. Draw a map giving instructions to your friend so that she reaches your house without any difficulty.

Do it with your own imagination.

3. Draw a map of your school compound using proper scale and symbols for various features like play ground main building, garden etc.

Do it with your own imagination.

2. Draw a map of your class room using proper scale and symbols for different objects.

Do it yourself.

1. Look at the given map of a city.

     

    Answer the following.

        (e) Which is further south, the primary school or the Sr. Secondary School?

        

(e) Sr. Secondary School is further south.

1. Look at the given map of a city.

    

    Answer the following.

        (d) Which is further east, the city park or the market?        

(d) The city park is further east.
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