Selina Textbook Solutions: Representing 3-D in 2-D | Mathematics Class 8 ICSE PDF Download

 Page 1


19. Representing 3-D in 2-D 
EXERCISE 19 
Question 1. 
If a polyhedron has 8 faces and 8 vertices, find the number of edges in it. 
Solution: 
Faces = 8 
Vertices = 8 
using Eulers formula, 
F + V – E = 2 
8 + 8 – E = 2 
-E = 2 – 16 
E= 14 
Question 2. 
If a polyhedron has 10 vertices and 7 faces, find the number of edges in it. 
Solution: 
Vertices = 10 
Faces = 7 
Using Eulers formula, 
F + V – E = 2 
7 + 10 – E = 2 
-E = -15 
E = 15 
Question 3. 
State, the number of faces, number of vertices and number of edges of: 
(i) a pentagonal pyramid 
(ii) a hexagonal prism 
Solution: 
(i) A pentagonal pyramid 
Number of faces = 6 
Number of vertices = 6 
Number of edges = 10 
(ii) A hexagonal prism 
Number of faces = 8 
Number of vertices = 12 
Number of edges = 18 
Question 4. 
Verily Euler’s formula for the following three dimensional figures: 
Page 2


19. Representing 3-D in 2-D 
EXERCISE 19 
Question 1. 
If a polyhedron has 8 faces and 8 vertices, find the number of edges in it. 
Solution: 
Faces = 8 
Vertices = 8 
using Eulers formula, 
F + V – E = 2 
8 + 8 – E = 2 
-E = 2 – 16 
E= 14 
Question 2. 
If a polyhedron has 10 vertices and 7 faces, find the number of edges in it. 
Solution: 
Vertices = 10 
Faces = 7 
Using Eulers formula, 
F + V – E = 2 
7 + 10 – E = 2 
-E = -15 
E = 15 
Question 3. 
State, the number of faces, number of vertices and number of edges of: 
(i) a pentagonal pyramid 
(ii) a hexagonal prism 
Solution: 
(i) A pentagonal pyramid 
Number of faces = 6 
Number of vertices = 6 
Number of edges = 10 
(ii) A hexagonal prism 
Number of faces = 8 
Number of vertices = 12 
Number of edges = 18 
Question 4. 
Verily Euler’s formula for the following three dimensional figures: 
 
Solution: 
(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 = 2 Hence 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 = 2 Hence 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 = 2 Hence proved. 
Question 5. 
Can a polyhedron have 8 faces, 26 edges and 16 vertices? 
Solution: 
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. 
Page 3


19. Representing 3-D in 2-D 
EXERCISE 19 
Question 1. 
If a polyhedron has 8 faces and 8 vertices, find the number of edges in it. 
Solution: 
Faces = 8 
Vertices = 8 
using Eulers formula, 
F + V – E = 2 
8 + 8 – E = 2 
-E = 2 – 16 
E= 14 
Question 2. 
If a polyhedron has 10 vertices and 7 faces, find the number of edges in it. 
Solution: 
Vertices = 10 
Faces = 7 
Using Eulers formula, 
F + V – E = 2 
7 + 10 – E = 2 
-E = -15 
E = 15 
Question 3. 
State, the number of faces, number of vertices and number of edges of: 
(i) a pentagonal pyramid 
(ii) a hexagonal prism 
Solution: 
(i) A pentagonal pyramid 
Number of faces = 6 
Number of vertices = 6 
Number of edges = 10 
(ii) A hexagonal prism 
Number of faces = 8 
Number of vertices = 12 
Number of edges = 18 
Question 4. 
Verily Euler’s formula for the following three dimensional figures: 
 
Solution: 
(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 = 2 Hence 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 = 2 Hence 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 = 2 Hence proved. 
Question 5. 
Can a polyhedron have 8 faces, 26 edges and 16 vertices? 
Solution: 
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. 
Question 6. 
Can a polyhedron have: 
(i) 3 triangles only ? 
(ii) 4 triangles only ? 
(iii) a square and four triangles ? 
Solution: 
(i) No. 
(ii) Yes. 
(iii) Yes. 
Question 7. 
Using Euler’s formula, find the values of x, y, z. 
 
Solution: 
 
Question 8. 
What is the least number of planes that can enclose a solid? What is the name of the 
solid. 
Solution: 
The least number of planes that can enclose a solid is 4. 
The name of the solid is Tetrahedron. 
Question 9. 
Is a square prism same as a cube? 
Solution: 
Yes, a square prism is same as a cube. 
Page 4


19. Representing 3-D in 2-D 
EXERCISE 19 
Question 1. 
If a polyhedron has 8 faces and 8 vertices, find the number of edges in it. 
Solution: 
Faces = 8 
Vertices = 8 
using Eulers formula, 
F + V – E = 2 
8 + 8 – E = 2 
-E = 2 – 16 
E= 14 
Question 2. 
If a polyhedron has 10 vertices and 7 faces, find the number of edges in it. 
Solution: 
Vertices = 10 
Faces = 7 
Using Eulers formula, 
F + V – E = 2 
7 + 10 – E = 2 
-E = -15 
E = 15 
Question 3. 
State, the number of faces, number of vertices and number of edges of: 
(i) a pentagonal pyramid 
(ii) a hexagonal prism 
Solution: 
(i) A pentagonal pyramid 
Number of faces = 6 
Number of vertices = 6 
Number of edges = 10 
(ii) A hexagonal prism 
Number of faces = 8 
Number of vertices = 12 
Number of edges = 18 
Question 4. 
Verily Euler’s formula for the following three dimensional figures: 
 
Solution: 
(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 = 2 Hence 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 = 2 Hence 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 = 2 Hence proved. 
Question 5. 
Can a polyhedron have 8 faces, 26 edges and 16 vertices? 
Solution: 
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. 
Question 6. 
Can a polyhedron have: 
(i) 3 triangles only ? 
(ii) 4 triangles only ? 
(iii) a square and four triangles ? 
Solution: 
(i) No. 
(ii) Yes. 
(iii) Yes. 
Question 7. 
Using Euler’s formula, find the values of x, y, z. 
 
Solution: 
 
Question 8. 
What is the least number of planes that can enclose a solid? What is the name of the 
solid. 
Solution: 
The least number of planes that can enclose a solid is 4. 
The name of the solid is Tetrahedron. 
Question 9. 
Is a square prism same as a cube? 
Solution: 
Yes, a square prism is same as a cube. 
Question 10. 
A cubical box is 6 cm x 4 cm x 2 cm. Draw two different nets of it. 
Solution: 
 
Question 11. 
Dice are cubes where the sum of the numbers on the opposite faces is 7. Find the 
missing numbers a, b and c. 
 
Solution: 
 
Question 12. 
Name the polyhedron that can be made by folding each of the following nets: 
 
Page 5


19. Representing 3-D in 2-D 
EXERCISE 19 
Question 1. 
If a polyhedron has 8 faces and 8 vertices, find the number of edges in it. 
Solution: 
Faces = 8 
Vertices = 8 
using Eulers formula, 
F + V – E = 2 
8 + 8 – E = 2 
-E = 2 – 16 
E= 14 
Question 2. 
If a polyhedron has 10 vertices and 7 faces, find the number of edges in it. 
Solution: 
Vertices = 10 
Faces = 7 
Using Eulers formula, 
F + V – E = 2 
7 + 10 – E = 2 
-E = -15 
E = 15 
Question 3. 
State, the number of faces, number of vertices and number of edges of: 
(i) a pentagonal pyramid 
(ii) a hexagonal prism 
Solution: 
(i) A pentagonal pyramid 
Number of faces = 6 
Number of vertices = 6 
Number of edges = 10 
(ii) A hexagonal prism 
Number of faces = 8 
Number of vertices = 12 
Number of edges = 18 
Question 4. 
Verily Euler’s formula for the following three dimensional figures: 
 
Solution: 
(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 = 2 Hence 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 = 2 Hence 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 = 2 Hence proved. 
Question 5. 
Can a polyhedron have 8 faces, 26 edges and 16 vertices? 
Solution: 
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. 
Question 6. 
Can a polyhedron have: 
(i) 3 triangles only ? 
(ii) 4 triangles only ? 
(iii) a square and four triangles ? 
Solution: 
(i) No. 
(ii) Yes. 
(iii) Yes. 
Question 7. 
Using Euler’s formula, find the values of x, y, z. 
 
Solution: 
 
Question 8. 
What is the least number of planes that can enclose a solid? What is the name of the 
solid. 
Solution: 
The least number of planes that can enclose a solid is 4. 
The name of the solid is Tetrahedron. 
Question 9. 
Is a square prism same as a cube? 
Solution: 
Yes, a square prism is same as a cube. 
Question 10. 
A cubical box is 6 cm x 4 cm x 2 cm. Draw two different nets of it. 
Solution: 
 
Question 11. 
Dice are cubes where the sum of the numbers on the opposite faces is 7. Find the 
missing numbers a, b and c. 
 
Solution: 
 
Question 12. 
Name the polyhedron that can be made by folding each of the following nets: 
 
Solution: 
(i) Triangular prism. It has 3 rectangles and 2 triangles. 
(ii) Triangular prism. It has 3 rectangles and 2 triangles. 
(iii) Hexagonal pyramid as it has a hexagonal base and 6 triangles. 
Question 13. 
Draw nets for the following polyhedrons: 
 
Solution: 
Net of hexagonal prism: 
 
Net of pentagonal pyramid: