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Question

Which of the following cannot be true for a polyhedron?

(a) V=4,F=4,E=6

(b) V=6,F=8,E=12

(c) V=20,F=12,E=30

(d) V=4,F=6,E=6


A

(a)

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B

(b)

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C

(c)

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D

(d)

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Solution

The correct option is D

(d)


Explanation for the correct option:

Option (d): Determine the polyhedron that cannot hold true for the given value.

Euler’s formula for any polyhedron is given by FaceF+VerticesV-EdgeE=2.

Consider V=4,F=6,E=6.

Substitute the above values into the formula.

FaceF+VerticesV-EdgeE=26+4-6=242

Hence, option (d) is correct.

Explanation for the incorrect options:

Option (a): Consider V=4,F=4,E=6.

Substitute the above values into the formula.

FaceF+VerticesV-EdgeE=24+4-6=22=2

Hence, option (a) is incorrect.

Option (b): Consider V=6,F=8,E=12.

Substitute the above values into the formula.

FaceF+VerticesV-EdgeE=28+6-12=22=2

Hence, option (b) is incorrect.

Option (d): Consider V=20,F=12,E=30.

Substitute the above values into the formula.

FaceF+VerticesV-EdgeE=212+20-30=22=2

Hence, option (c) is incorrect.

Therefore, options (a), (b), and (c) are incorrect.

Hence, option (d) is correct.


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