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\ 06-Jan-2025 Surendra Prajapati Proof-7 Reader’s Sign _______________________ Date __________

3. The number of edges in an octahedron is 12.

4. Euler’s formula is true for all 3-D shapes.
5. Total number of regular convex polyhedrons is 5.
E. Answer the following.

1. Can a polyhedron has V = F = 9 and E = 16?
2. A polyhedron has 12 faces and 20 vertices, find the edges.
3. A solid has 60 edges and 40 faces. How many vertices will it have?

4. Using Euler’s formula, find the values of X, Y, Z, P and Q in the given table.
Faces 7 Y 9 P 8

Vertices 10 12 Z 6 Q
Edges X 18 16 12 12

5. Draw a prism with its base as regular hexagon with one of its face facing you. Now, draw the top
view, front view and side view of the solid.
6. When the Maths teacher explained the topic ‘Visualising Solid Shapes’, she wants the students
to memorise the formula showing the relation among number of faces (F), number of vertices
(V) and number of edges (E). She asked in class how we call number ‘5’ in words. All students Case study
replied in unison–‘FiVE’.

Great! said the teacher. Then she again asked which digit resembles the alphabet ‘i’? All students
uttered with vigour–one (1).
Nice! what comes after one ‘1’? The whole class replied–‘Two (2)’.
Yeah!—You all have done good job. All my students are genius like ‘Euler’. She gave compliments
to her students. The students replied with naughty smiles on their faces, that ‘EulEr’ also
have ‘FiVE’ letters. The teacher further said as ‘i’ is a symbol of ‘Ego’, so we remove i from the
spelling of FiVE, to get ‘FVE’.
Here in the class we are ‘2’, i (teacher) and you (student) so add ‘2’ in the term ‘FVE’ with an
equal sign in the middle. What we get!
F + V = E + 2
Congrats! You have developed a formula my little genius.

Now, answer the following questions:
( a) if a polyhedron has 4 faces, and 4 vertices, then find the number of edges.

( b) if for a polyhedron V = 7 and E = 12, then find the value of ‘F’.
( c) Which solid has the following: F = 6, V = 8, E = 12?
( d) if a polyhedron has 20 faces and 30 edges, then how many vertices does it have?

create and solve
With the help of internet, teachers or parents, draw the nets of platonic solids on a thick paper/
cardboard. Cut out the layout, fold and glue/tape to make a solid shape. Finally, verify the Euler’s
formula.

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