Visualising Shapes
Class 8th Mathematics RD Sharma Solution
- What is the least number of planes that can enclose a solid? What is the name…
- Can a polyhedron have for its faces? (i) 3 triangles? (ii) 4 triangles? (iii) a…
- Is it possible to have a polyhedron with any given number of faces?…
- Is a square prism same as a cube?
- Can a polyhedron have 10 faces, 20 edges and 15 vertices?
- Verify Eulers formula for each of the following polyhedrons:
- Using Eulers formula find the unknown: Faces ? 5 20 Vertices 6 ? 12 Edges 12 9…
Exercise 19.1
Question 1.What is the least number of planes that can enclose a solid? What is the name of the solid?
Answer:
Four planes are required to enclose a solid.
The name of solid is tretrahedron.
Question 2.
Can a polyhedron have for its faces?
(i) 3 triangles?
(ii) 4 triangles?
(iii) a square and four triangles?
Answer:
(i) 3 triangles?
No, Because a polyhedron is a solid shape bounded by polygons.
(ii) 4 triangles?
Yes, Because four triangles will form a tetrahedron, which is a polygon.
(iii) a square and four triangles?
Yes, because a square pyramid has a square and four triangles as its faces. Since pyramid is a polyhedron whose base is a polygon of any number of sides and whose other faces are triangles with common vertex.
Question 3.
Is it possible to have a polyhedron with any given number of faces?
Answer:
Yes, if number of faces is four or more.
For example pyramid is a polyhedron whose base is a polygon of any number of sides and whose other faces are triangles with common vertex.
Question 4.
Is a square prism same as a cube?
Answer:
Yes, a square is a three dimensional shape with six rectangular shaped sides, at least two of which are squares. Cubes are rectangular prisms length, width and height of same measurement.
Question 5.
Can a polyhedron have 10 faces, 20 edges and 15 vertices?
Answer:
No,
Using Euler’s formula
V + F = E + 2
15 + 10 = 20 + 2
25 ≠ 22
Since the given polyhedron is not following Euler’s formula, therefore its not possible.
Question 6.
Verify Euler’s formula for each of the following polyhedrons:
Answer:
(i) Vertices = 10
Faces = 7
Edges = 15
V + F = E + 2
10 + 7 = 15 + 2
17 = 17
(ii) Vertices = 9
Faces = 9
Edges = 16
V + F = E + 2
9 + 9 = 16 + 2
18 = 18
(iii) Vertices = 14
Faces = 8
Edges = 20
V + F = E + 2
14 + 8 = 20 + 2
22 = 22
(iv) Vertices = 6
Faces = 8
Edges = 12
V + F = E + 2
6 + 8 = 12 + 2
14 = 14
(v) Vertices = 9
Faces = 9
Edges = 16
V + F = E + 2
9 + 9 = 16 + 2
18 = 18
Question 7.
Using Euler’s formula find the unknown:
Answer:
(i) V + F = E + 2
6 + F = 12 + 2
F = 14 - 6
F = 8
Therefore number of faces are 8
(ii) V + F = E + 2
V + 5 = 9 + 2
V = 11 - 5
V = 6
Therefore number of vertices are 6
(iii) V + F = E + 2
12 + 20 = E + 2
E = 32 - 2
E = 30
Therefore number of edges are 8
Exercise 19.2
Question 1.Which among of the following are nets for a cube?
Answer:
Figure (iii) and (vi) are the nets of a cube.
Question 2.
Name the polyhedron that can be made by folding each net:
Answer:
(i) From first figure Square pyramid can be made
(ii) From second figure Triangular prism can be made
(iii) From third figure Triangular prism can be made
(iv) From fourth figure Hexagonal prism can be made
(iv) From fifth figure Hexagonal pyramid can be made
(v) From fifth figure Cube can be made
Question 3.
Dice are cubes where the numbers on the opposite faces must total 7. Which of the following are dice?
Answer:
Fig (i) is a dice because the sum of numbers on opposite faces is 7 (3 + 4 = 7 and 6 + 1 = 7).
Question 4.
Draw nets for each of the following polyhedrons:
Answer:
(i) Net pattern of a cube:
(ii) Net pattern of Triangular prism:
(iii) Net pattern of Hexagonal prism:
Question 5.
Match the following figures:
Answer:
(a)—(iv) Because multiplication of numbers on adjacent faces are equal, i.e 6×4 = 24 and 4×4 = 16
(b)—(i) Because multiplication of numbers on adjacent faces are equal, i.e 3×3 = 9 and 8×3 = 24
(c)—(ii) Because multiplication of numbers on adjacent faces are equal, i.e 6×4 = 24 and 6×3 = 18
(d)—(iii) Because multiplication of numbers on adjacent faces are equal, i.e 3×3 = 9 and 3×9 = 27