Class 8 Mathematics Notes Chapter 10 (Visualising Solid Shapes) – Mathematics Book

Alright class, let's focus on Chapter 10: Visualising Solid Shapes. This chapter is fundamental for developing spatial reasoning skills, which can be surprisingly useful in various sections of government exams, especially those testing logical or quantitative aptitude. We'll break down the key concepts precisely.

Chapter 10: Visualising Solid Shapes - Detailed Notes

1. Introduction: Plane Shapes vs. Solid Shapes

• Plane Shapes (2-Dimensional or 2D):
  • Have only two measurements: length and breadth.
  • Can be drawn completely on a flat surface (plane).
  • Examples: Square, Rectangle, Circle, Triangle.
  • They have area and perimeter, but no volume.
• Solid Shapes (3-Dimensional or 3D):
  • Have three measurements: length, breadth, and height (or depth).
  • Occupy space.
  • Cannot be drawn completely on a flat surface (though we can represent them).
  • Examples: Cube, Cuboid, Sphere, Cylinder, Cone, Pyramid.
  • They have surface area and volume.

2. Views of 3D Shapes

• A 3D object looks different from different positions. The standard views are:
  • Top View: How the object looks when viewed directly from above.
  • Front View: How the object looks when viewed directly from the front.
  • Side View: How the object looks when viewed directly from the side (usually the left or right side is specified or implied).
• Importance: Understanding these views helps interpret technical drawings, maps, and diagrams, and develops spatial visualization.
  • Example: For a simple house, the front view might show the door and front windows, the side view might show side windows, and the top view might show the roof's shape.

3. Mapping Space Around Us (Brief Overview)

• Maps are used to locate places. They use:
  • Scale: A ratio representing the relationship between distance on the map and actual distance on the ground.
  • Symbols: Standard signs representing different features (rivers, roads, buildings, etc.).
  • Perspective: Maps depict a top-down view, but relative positions and distances are key.

4. Faces, Edges, and Vertices

• These are fundamental components of many solid shapes, particularly polyhedrons.

  • Faces (F): The flat surfaces of a solid shape. These are polygons (like squares, triangles, rectangles).
  • Edges (E): The line segments where two faces meet.
  • Vertices (V): The points (corners) where three or more edges meet.

• Examples:

  • Cube: 6 Faces (squares), 12 Edges, 8 Vertices.
  • Cuboid: 6 Faces (rectangles), 12 Edges, 8 Vertices.
  • Square Pyramid: 5 Faces (1 square base, 4 triangular sides), 8 Edges, 5 Vertices.
  • Triangular Prism: 5 Faces (2 triangular bases, 3 rectangular sides), 9 Edges, 6 Vertices.

5. Polyhedrons

• Definition: A solid shape whose surfaces are flat polygonal faces, edges are straight line segments, and vertices are sharp corners.
• Types:
  • Convex Polyhedron: A polyhedron where any line segment connecting two points on its surface lies entirely inside or on the polyhedron. (All examples above are convex).
  • Regular Polyhedron (Platonic Solids): A polyhedron whose faces are congruent regular polygons, and the same number of faces meet at each vertex. (Examples: Cube - faces are squares, 3 meet at each vertex; Tetrahedron - faces are equilateral triangles, 3 meet at each vertex).
• Prisms: Polyhedrons with two identical and parallel polygonal bases, and rectangular side faces connecting corresponding edges of the bases. (Named after the shape of the base: Triangular Prism, Square Prism (Cube/Cuboid), Pentagonal Prism, etc.)
• Pyramids: Polyhedrons with a polygonal base and triangular side faces that meet at a common vertex (apex). (Named after the shape of the base: Triangular Pyramid (Tetrahedron), Square Pyramid, Pentagonal Pyramid, etc.)

6. Non-Polyhedrons

• Solid shapes that have curved surfaces are not polyhedrons.
• Examples:
  • Cylinder: Two parallel circular bases and one curved surface.
  • Cone: One circular base, one vertex (apex), and one curved surface.
  • Sphere: A perfectly round shape with every point on its surface equidistant from the center. Has only one curved surface.

7. Euler's Formula (VERY IMPORTANT)

• For any polyhedron, the relationship between the number of Faces (F), Vertices (V), and Edges (E) is given by:
  F + V - E = 2
• Application:
  • Can be used to verify if a given shape is a polyhedron (if the formula holds).
  • If two of F, V, or E are known for a polyhedron, the third can be calculated.
• Verification Examples:
  • Cube: F=6, V=8, E=12. So, 6 + 8 - 12 = 14 - 12 = 2. (Holds)
  • Square Pyramid: F=5, V=5, E=8. So, 5 + 5 - 8 = 10 - 8 = 2. (Holds)
  • Triangular Prism: F=5, V=6, E=9. So, 5 + 6 - 9 = 11 - 9 = 2. (Holds)
• Limitation: Euler's formula does not apply to non-polyhedrons (like spheres, cones, cylinders).

8. Nets for Building 3D Shapes

• Definition: A net is a 2D pattern that can be folded along its edges to form a 3D shape.
• Importance: Helps understand the relationship between 2D representations and 3D objects. Visualizing how a shape unfolds or folds is key.
• Common Nets: Be familiar with nets for cubes, cuboids, cylinders, cones, pyramids, and prisms. Note that a single solid shape can often have multiple different nets.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed. These are typical of the foundational understanding required for competitive exams.

1. Which of the following is a 3-Dimensional shape?
(A) Circle
(B) Square
(C) Triangle
(D) Sphere

2. A solid shape has 6 faces, 12 edges, and 8 vertices. What is this shape likely to be?
(A) Square Pyramid
(B) Triangular Prism
(C) Cube
(D) Cone

3. What is the top view of a standard dice (showing '1' on top, '2' on front, '3' on right)?
(A) A square with 6 dots
(B) A square with 1 dot
(C) A square with 2 dots
(D) A square with 3 dots

4. Which formula correctly represents Euler's formula for polyhedrons?
(A) F + E - V = 2
(B) V + E - F = 2
(C) F + V - E = 2
(D) F - V + E = 2

5. A polyhedron has 7 faces and 10 vertices. How many edges does it have?
(A) 15
(B) 17
(C) 5
(D) 19

6. Which of the following is NOT a polyhedron?
(A) Cuboid
(B) Cylinder
(C) Triangular Pyramid
(D) Pentagonal Prism

7. How many faces does a hexagonal pyramid have?
(A) 6
(B) 7
(C) 8
(D) 12

8. A net for which 3D shape will typically consist of one rectangle and two circles?
(A) Cone
(B) Cube
(C) Cylinder
(D) Sphere

9. What is the shape of the side faces of any prism?
(A) Triangle
(B) Rectangle
(C) Circle
(D) Depends on the base shape

10. If you view a cone from directly above (top view), what shape do you see (assuming the base is visible)?
(A) A triangle
(B) A circle with a point in the center
(C) A rectangle
(D) A circle

Answer Key for MCQs:

1. (D) Sphere (Others are 2D)
2. (C) Cube (Matches the F, V, E count for a cube/cuboid)
3. (B) A square with 1 dot (Top view shows the top face)
4. (C) F + V - E = 2 (Standard Euler's formula)
5. (A) 15 (Using F + V - E = 2 => 7 + 10 - E = 2 => 17 - E = 2 => E = 15)
6. (B) Cylinder (It has a curved surface)
7. (B) 7 (1 hexagonal base + 6 triangular side faces)
8. (C) Cylinder (The rectangle forms the curved surface, the circles are the bases)
9. (B) Rectangle (By definition of a prism)
10. (B) A circle with a point in the center (The circle is the base, the point is the apex)

Study these notes carefully. Focus on understanding the definitions, properties, and especially Euler's formula. Practice identifying faces, edges, vertices, and visualizing the different views and nets. Good luck with your preparation!