CHAPTER 19 - REPRESENTING 3-D IN 2-D - Q&A
EXERCISE (Page 234)
8. What is the least number of planes that can enclose a solid? What is the name of the solid?
Answer:
The least number of planes required to enclose a solid is 4.
The name of the solid is a Tetrahedron (or Triangular Pyramid).
Explanation: A solid must be closed. Three planes would form an open prism-like shape or infinite space. Four planes can meet to form a closed pyramid with a triangular base.
9. Is a square prism same as a cube?
Answer:
No.
Explanation: A cube is a special type of square prism where all faces are squares (height = side of base). A square prism has a square base, but its height can be different from the side of the base, resulting in rectangular lateral faces.
2. If a polyhedron has 10 vertices and 7 faces, find the number of edges in it.
Answer:
Using Euler's Formula: F + V - E = 2
Given: F = 7, V = 10
7 + 10 - E = 2
17 - E = 2
E = 17 - 2
E = 15
So, the number of edges is 15.
3. State the number of faces, number of vertices and the number of edges of:
(i) a pentagonal pyramid
(ii) a hexagonal prism
Answer:
(i) Pentagonal Pyramid:
Base: Pentagon (5 sides)
Faces (F) = 1 (base) + 5 (triangular faces) = 6
Vertices (V) = 5 (base corners) + 1 (apex) = 6
Edges (E) = 5 (base edges) + 5 (slant edges) = 10
(ii) Hexagonal Prism:
Base: Hexagon (6 sides)
Faces (F) = 2 (base and top) + 6 (rectangular faces) = 8
Vertices (V) = 2 × number of sides = 2 × 6 = 12
Edges (E) = 3 × number of sides = 3 × 6 = 18
4. Verify Euler's formula for the following three dimensional figures.
Answer:
(Assuming the figures refer to standard shapes like the ones calculated above, such as a Pentagonal Pyramid and Hexagonal Prism)
For Pentagonal Pyramid:
F = 6, V = 6, E = 10
F + V - E = 6 + 6 - 10 = 12 - 10 = 2. (Verified)
For Hexagonal Prism:
F = 8, V = 12, E = 18
F + V - E = 8 + 12 - 18 = 20 - 18 = 2. (Verified)
10. The dimensions of a cuboid are 6 cm x 4 cm x 2 cm. Draw two different nets of it.
Answer:
To draw the nets, visualize unfolding the cuboid. It has 3 pairs of rectangular faces: 6x4, 6x2, and 4x2.
Net 1 (Cross shape): Align four faces in a row (e.g., 6x4, 6x2, 6x4, 6x2) and attach the two remaining faces (4x2) to the sides of one of the 6x4 faces.
Net 2 (T-shape): Arrange the 6x4, 6x2, 6x4 faces in a column and attach the side faces (4x2) and the top/bottom faces appropriately to close the shape.
(Note: This requires drawing on paper based on the dimensions.)
11. Dice are cubes where the sum of the numbers on the opposite faces is 7. Find the missing numbers a, b and c.
Answer:
In a standard die, opposite faces sum to 7.
Pairs are: (1, 6), (2, 5), (3, 4).
Without the specific image layout, we apply the rule:
- If 'a' is opposite a known number, subtract that number from 7 to find 'a'.
- Example: If 'a' is opposite 5, then a = 7 - 5 = 2.
- Example: If 'b' is opposite 6, then b = 7 - 6 = 1.
- Example: If 'c' is opposite 4, then c = 7 - 4 = 3.
TEST YOURSELF (Page 235-236)
1. Multiple Choice Type: Choose the correct answer from the options given below.
(i) If in a polyhedron, number of faces = 20 and number edges = 30; the number of vertices is:
(a) 12
(b) 6
(c) 8
(d) 20
Answer: (a) 12
Explanation: Using Euler’s Formula F + V - E = 2.
20 + V - 30 = 2
V - 10 = 2
V = 12.
(iii) Joseph is making a pentagonal prism using identical straws. How many straws does he need?
(a) 20
(b) 15
(c) 10
(d) 30
Answer: (b) 15
Explanation: A pentagonal prism has 2 pentagonal bases and 5 rectangular sides.
Number of edges (straws) = 5 (bottom base) + 5 (top base) + 5 (vertical edges) = 15.
The following questions are Assertion-Reason based questions. Choose your answer based on the codes given below.
(1) Both A and R are correct, and R is the correct explanation for A.
(2) Both A and R are correct, and R is not the correct explanation for A.
(3) A is true, but R is false.
(4) A is false, but R is true.
(vii) Assertion (A): In a polyhedron, if there are 6 vertices, 12 edges, then the number of faces are 8.
Reason (R): In a pentagonal pyramid, there are 6 faces, 6 vertices and 10 edges.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (b) (2)
Explanation:
Check A: F + V - E = 2 => F + 6 - 12 = 2 => F - 6 = 2 => F = 8. (True).
Check R: Pentagonal pyramid has 6 faces, 6 vertices, 10 edges. (True).
Relation: R is a specific example and does not explain the general formula used in A. Thus, both are true but R is not the explanation for A.
(viii) Assertion (A): If a polyhedron has 7 vertices and 10 faces, the number of edges is 19.
Reason (R): The relationship between the faces (F), edges (E) and vertices (V) of a polyhedron is F + V - E = 2.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (d) (4)
Explanation:
Check A: Using formula F + V - E = 2 => 10 + 7 - E = 2 => 17 - E = 2 => E = 15. The assertion says 19, which is False.
Check R: The formula given is correct. (True).
Conclusion: A is false, R is true.
(ix) Assertion (A): The number of edges in a triangular prism = 9.
Reason (R): In a triangular prism, the number of vertices = 2 x number of sides = 6; The number of faces = 2 + number of sides = 5.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (a) (1)
Explanation:
Check A: Triangular prism has 9 edges. (True).
Check R: Vertices = 2*3 = 6. Faces = 2+3 = 5. (True).
Relation: Using the values from R (V=6, F=5) in Euler's formula (5+6-E=2) gives E=9, which is the statement in A. R provides the correct components to derive A.
(x) Assertion (A): The number of edges in a rectangular pyramid = 8.
Reason (R): In a triangular prism, the number of vertices is one more than the number of sides and the number of faces is one less than the number of sides.
(a) (1)
(b) (2)
(c) (3)
(d) (4)
Answer: (c) (3)
Explanation:
Check A: Rectangular pyramid has 4 base edges + 4 slant edges = 8 edges. (True).
Check R: For a triangular prism (sides=3), Vertices should be 6, but "one more than sides" is 4. Faces should be 5, but "one less than sides" is 2. This statement is False.
Conclusion: A is true, R is false.
8. Name the polyhedron that can be made by folding each of the following nets:
(i) [Net with 3 rectangles and 2 triangles]
(ii) [Net with rectangles and triangles]
Answer:
(i) Triangular Prism
(ii) Triangular Prism (or dependent on the specific image, likely a variation of a prism or pyramid).