RD Sharma solution class 8 chapter 19 Visualising Shapes Exercise 19.1

Exercise 19.1

Page-19.9

Question 1:

What is the least number of planes that can enclose a solid? What is the name of the solid?

Answer 1:

$$\begin{gathered} \text{The least number of planes that can enclose a solid is 4.} \\ \text{Tetrahedron is a solid with four planes (faces).} \end{gathered}$$

Question 2:

Can a polyhedron have for its faces:
(i) 3 triangles?
(ii) 4 triangles?
(iii) a square and four triangles?

Answer 2:

$$\begin{gathered} \text{(i)} \\ \text{No, because in order to complete a polyhedron, we need at least four triangular faces. } \\ \text{(ii)} \\ \text{Yes, a polyhedron with 4 triangular faces is a tetrahedron.} \\ \text{(iii)} \\ \text{Yes, with the help of a square bottom and four triangle faces, we can form a pyramid.} \end{gathered}$$

Question 3:

Is it possible to have a polyhedron with any given number of faces?

Answer 3:

$$\begin{gathered} \text{Yes, it is possible to have a polyhedron with any number of faces. } \\ \text{The only condition is that there should be at least four faces. } \\ \text{This is because there is no possible polyhedron with 3 or less faces.} \end{gathered}$$

Question 4:

Is a square prism same as a cube?

Answer 4:

$$\begin{gathered} \text{Yes, a square prism and a cube are the same.} \\ \text{Both of them have 6 faces, 8 vertices and 12 edges.} \\ \text{The only difference is that a cube has 6 equal faces, while a square prism has a shape like a cuboid with two square} \\ \text{faces, one at the top and the other at the bottom and with, possibly, 4 rectangular faces in between. } \end{gathered}$$

Question 5:

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

Answer 5:

$$\begin{gathered} \text{No, because every polyhedron satisfies Euler's formula, given below:} \\ \text{ F+V=E+2} \\ \text{Here, number of faces F = 10} \\ \text{Number of edges E = 20} \\ \mathrm{N}\text{umber of vertices V = 15} \\ \text{So, by Euler's formula:} \\ \mathrm{LHS}: \text{10+15 = 25} \\ \mathrm{RHS}: 20+2 = 22, \\ \text{which is not true because 25≠22} \\ \mathrm{Hence}, \mathrm{Eulers} \mathrm{formula} \mathrm{is} \mathrm{not} \mathrm{satisfied} \mathrm{and} \mathrm{no} \mathrm{polyhedron} \mathrm{may} \mathrm{be} \mathrm{formed}. \end{gathered}$$

Question 6:

Verify Euler's formula for each of the following polyhedrons:

Answer 6:

$$\begin{gathered} \text{(i)} \\ \text{In the given polyhedron:} \\ \text{Edges E=15} \\ \text{Faces F=7} \\ \text{Vertices V=10} \end{gathered}$$


$$\begin{gathered} \text{Now, putting these values in Euler's formula:} \\ \mathrm{LHS}: \text{F+V} \\ = \text{7+10} \\ = \text{17} \\ \mathrm{LHS}: \mathrm{E}+2 \\ =15+2 \\ =17 \\ \mathrm{LHS} = \mathrm{RHS} \\ \text{Hence, the Euler's formula is satisfied.} \end{gathered}$$

$$\begin{gathered} \text{(ii)} \\ \text{In the given polyhedron:} \\ \text{Edges E=16} \\ \text{Faces F=9} \\ \text{Vertices V=9} \end{gathered}$$


$$\begin{gathered} \text{Now, putting these values in Euler's formula:} \\ \mathrm{RHS}: \mathrm{F}\text{+V} \\ = 9\text{+9} \\ = \text{18} \\ \mathrm{LHS}: \mathrm{E}+2 \\ =16+2 \\ =18 \\ \text{LHS = RHS} \\ \text{Hence, Euler's formula is satisfied.} \end{gathered}$$

$$\begin{gathered} \text{(iii)} \\ \text{In the following polyhedron:} \\ \text{Edges E=21} \\ \text{Faces F=9} \\ \text{Vertices V=14} \end{gathered}$$

$$\begin{gathered} \text{Now, putting these values in Euler's formula:} \\ LHS: \text{F+V} \\ = \text{9+14} \\ = \text{23} \\ RHS: \mathrm{E+2} \\ =21+2 \\ =23 \\ \mathrm{This}\text{ is true. } \\ \text{Hence, Euler's formula is satisfied.} \end{gathered}$$

$$\begin{gathered} \text{(iv)} \\ \text{In the following polyhedron:} \\ \text{Edges E=8} \\ \text{Faces F=5} \\ \text{Vertices V=5} \end{gathered}$$

$$\begin{gathered} \text{Now, putting these values in Euler's formula:} \\ \mathrm{LHS}: \text{F+V} \\ = 5+5 \\ = 10 \\ \mathrm{RHS}: \mathrm{E}+2 \\ \text{=8+2} \\ \text{=10} \\ \text{LHS = RHS} \\ \text{Hence, Euler's formula is satisfied.} \end{gathered}$$

$$\begin{gathered} \text{(v)} \\ \text{In the following polyhedron:} \\ \text{Edges E=16} \\ \text{Faces F=9} \\ \text{Vertices V=9} \end{gathered}$$

$$\begin{gathered} \text{Now, putting these values in Euler's formula:} \\ \mathrm{LHS}: \text{F+V} \\ = 9+9 \\ = 18 \\ \mathrm{RHS}: \mathrm{E}+2 \\ \text{=16+2} \\ \text{=18} \\ \text{LHS = RHS} \\ \text{Hence, Euler's formula is satisfied.} \end{gathered}$$

Page-19.10

Question 7:

Using Euler's formula find the unknown:

Faces	?	5	20
Vertices	6	?	12
Edges	12	9	?

Answer 7:

$$\begin{gathered} \text{We know that the Euler's formula is: F+V = E+2} \\ \text{(i) } \\ \text{The number of vertices V is 6 and the number of edges E is 12.} \\ \text{Using Euler's formula:} \\ \text{F+6 = 12+2} \\ \text{F+6 = 14} \\ \text{F = 14-6} \\ \text{F = 8} \\ \text{So, the number of faces in this polyhedron is 8.} \\ \text{(ii)} \\ \text{Faces, F = 5} \\ \mathrm{E}\text{dges, E = 9.} \\ \mathrm{W}\text{e have to find the number of vertices.} \\ \text{Putting these values in Euler's formula:} \\ \text{5+V = 9+2} \\ \text{5+V = 11} \\ \text{V = 11-5} \\ \text{V = 6} \\ \text{So, the number of vertices in this polyhedron is 6.} \\ \text{(iii)} \\ \mathrm{N}\text{umber of faces F = 20} \\ \mathrm{N}\text{umber of vertices V = 12} \\ \text{Using Euler's formula:} \\ \text{20+12 = E+2} \\ \text{32 = E+2} \\ \text{E+2 = 32} \\ \text{E = 32-2} \\ \text{E = 30.} \\ \text{So, the number of edges in this polyhedron is 30.} \end{gathered}$$