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ICSE-VIII-Mathematics

19: Representing 3-D in 2-D Class 8 Maths

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  • #
    Exercise 19
  • Qstn #1
    If a polyhedron has 8 faces and 8 vertices, find the number of edges in it.
    Ans : Faces = 8
    Vertices = 8
    using Eulers formula,
    F + V - E = 2
    ⇒ 8 + 8 - E = 2
    ⇒ -E = 2 - 16
    ⇒ E= 14
  • Qstn #2
    If a polyhedron has 10 vertices and 7 faces, find the number of edges in it.
    Ans : Vertices = 10
    Faces = 7
    Using Eulers formula,
    F + V - E = 2
    ⇒ 7 + 10 - E = 2
    ⇒ -E = -15
    ⇒ E = 15
  • Qstn #3
    State, the number of faces, number of vertices and number of edges of:
  • #3-i
    (a) pentagonal pyramid
    Ans : A pentagonal pyramid
    Number of faces = 6
    Number of vertices = 6
    Number of edges = 10
  • #3-ii
    (a) hexagonal prism
    Ans : A hexagonal prism
    Number of faces = 8
    Number of vertices = 12
    Number of edges = 18
  • Qstn #4
    Verily Euler’s formula for the following three dimensional figures:
    Ans : (i) Number of vertices = 6
    Number of faces = 8
    Number of edges = 12
    Using Euler formula,
    F + V - E = 2
    ⇒ 8 + 6 - 12 = 2
    ⇒ 2 = 2Hence, proved.
    (ii) Number of vertices = 9
    Number of faces = 8
    Number of edges = 15
    Using, Euler’s formula,
    F + V - E = 2
    ⇒ 9 + 8 - 15 = 2
    ⇒ 2 = 2Hence, proved.
    (iii) Number of vertices = 9
    Number of faces = 5
    Number of edges = 12
    Using, Euler’s formula,
    F + V - E = 2
    ⇒ 9 + 5 - 12 = 2
    ⇒ 2 = 2Hence, proved.
  • Qstn #5
    Can a polyhedron have 8 faces, 26 edges and 16 vertices?
    Ans : Number of faces = 8
    Number of vertices = 16
    Number of edges = 26
    Using Euler’s formula
    F + V - E
    ⇒ 8 + 16 - 26 ≠ -2
    ⇒ -2 ≠ 2
    No, a polyhedron cannot have 8 faces, 26 edges and 16 vertices.
  • #6-i
    3 triangles only?
    Ans : No
  • #6-ii
    4 triangles only?
    Ans : Yes
  • #6-iii
    (a) square and four triangles?
    Ans : Yes
  • Qstn #7
    Using Euler’s formula, find the values of x, y, z.

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    Faces
    Vertices
    Edges
  • #7-i
    x
    15
    20
    Ans : F + V - E = 2
    ⇒ x + 15 - 20 = 2
    ⇒ x - 5 = 2
    ⇒ = 2 + 5 = 7
  • #7-ii
    6
    y
    8
    Ans : F + V - E = 2
    ⇒ 15 + y - 26 = 2
    ⇒ y - 11 = 2
    ⇒ y = 2 + 11
    ⇒ y = 13
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