Class 7 Notes

Class 7 Mathematics Notes Chapter 15 (Visualising Solid Shapes) – Mathematics Book

Philoid

Philoid

— 6 min read

Mathematics
Detailed Notes with MCQs of Chapter 15: Visualising Solid Shapes. This is a crucial chapter not just for your current class, but it builds a foundation for geometry and spatial reasoning often tested in various government exams. We need to understand how to represent and interpret three-dimensional objects in two dimensions.

Chapter 15: 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 (a plane).
    • Examples: Square, Rectangle, Circle, Triangle.
    • They have area 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 while showing all features accurately without specific techniques.
    • Examples: Cube, Cuboid, Sphere, Cylinder, Cone, Pyramid.
    • They have surface area and volume.

2. Elements of 3D Shapes (Especially Polyhedrons)

Many solid shapes are made up of polygonal regions. These are called Polyhedrons (singular: Polyhedron). Examples: Cube, Cuboid, Prisms, Pyramids. Shapes like Cylinders, Cones, Spheres are not polyhedrons because they have curved surfaces.

For polyhedrons, we identify three key elements:

  • Faces (F): The flat surfaces of the solid shape. These are polygons.

    • Example: A cube has 6 square faces. A cuboid has 6 rectangular faces (can include squares). A triangular pyramid (tetrahedron) has 4 triangular faces.

  • Edges (E): The line segments where two faces meet.

    • Example: A cube has 12 edges. A cuboid has 12 edges. A triangular pyramid has 6 edges.

  • Vertices (V): The points where three or more edges meet (the corners).

    • Example: A cube has 8 vertices. A cuboid has 8 vertices. A triangular pyramid has 4 vertices.

  • Euler's Formula for Polyhedrons: A fundamental relationship connecting Faces (F), Vertices (V), and Edges (E) for any polyhedron:
    F + V - E = 2

    • Example (Cube): F=6, V=8, E=12. So, 6 + 8 - 12 = 14 - 12 = 2.
    • Example (Triangular Pyramid): F=4, V=4, E=6. So, 4 + 4 - 6 = 8 - 6 = 2.
    • This formula is a useful check and often appears in competitive exams.

3. Nets for Building 3D Shapes

  • Definition: A net is a 2-dimensional shape (a pattern or layout) that can be folded along its edges to form a 3-dimensional solid shape.
  • Purpose: Helps understand the relationship between 2D representations and 3D objects. Visualises how the faces of a solid fit together.
  • Examples:
    • Cube: A common net consists of six squares arranged in a cross shape (or other valid arrangements). There are 11 distinct nets for a cube.
    • Cuboid: Similar to a cube, but uses rectangles (and possibly squares).
    • Cylinder: A net consists of one rectangle (for the curved surface) and two circles (for the top and bottom bases).
    • Cone: A net consists of one sector of a circle (for the curved surface) and one circle (for the base).
    • Pyramid (Square Base): A net consists of one square (base) and four triangles attached to its sides.
  • Key Skill: Identifying whether a given 2D pattern can actually fold into a specific 3D shape.

4. Drawing Solids on a Flat Surface

Representing 3D shapes on 2D paper requires specific techniques:

  • Oblique Sketches:
    • Simple way to give a visual impression of a 3D object.
    • Easy to draw.
    • Characteristics:
      • The front face (and sometimes the opposite back face) is drawn true to shape and size.
      • Edges going back from the front face (receding lines) are drawn typically at a 45° angle to the horizontal.
      • Lengths of these receding edges are often drawn shorter than the actual proportional length to give a sense of perspective, but they are not accurately scaled.
      • Does not maintain accurate proportions or angles of the solid.
  • Isometric Sketches:
    • Provides a more accurate representation of the solid's dimensions and proportions.
    • Drawn on special isometric dot paper, which has dots arranged in equilateral triangles.
    • Characteristics:
      • Represents the object's length, width, and height accurately in proportion.
      • Vertical lines are drawn vertically.
      • Horizontal edges are drawn usually at a 30° angle to the horizontal baseline (following the dot grid).
      • All lines parallel to the three main axes (length, width, height) are drawn to scale.

5. Visualising Solid Objects - Different Ways

Understanding a 3D object involves being able to visualise it from different perspectives or imagine its internal structure.

  • (i) Slicing or Cutting (Cross-sections):
    • Imagine cutting through a solid shape with a straight cut (like with a knife). The shape of the flat surface exposed by the cut is called the cross-section.
    • The shape of the cross-section depends on:
      • The solid shape itself.
      • The angle/orientation of the cut (horizontal, vertical, slanted).
    • Examples:
      • Horizontal cut through a cylinder -> Circle.
      • Vertical cut through a cylinder -> Rectangle.
      • Vertical cut through a cone (through the apex) -> Triangle.
      • Horizontal cut through a cone -> Circle.
      • Cut through a cube parallel to a face -> Square.
      • Diagonal cut through a cube -> Can be a rectangle or other polygons depending on the cut.
  • (ii) Shadow Play:
    • A 3D object casts a 2D shadow when light shines on it.
    • The shape of the shadow depends on:
      • The solid shape.
      • The direction and type of the light source (e.g., torchlight overhead vs. from the side).
      • The orientation of the object relative to the light.
    • Examples (with light directly overhead):
      • Sphere -> Circle shadow.
      • Cube -> Square shadow.
      • Cone (standing on base) -> Circle shadow.
      • Cylinder (standing on base) -> Circle shadow.
    • Examples (with light from the front):
      • Cube -> Square shadow.
      • Cone (standing on base) -> Triangle shadow.
      • Cylinder (standing on base) -> Rectangle shadow.
  • (iii) Viewing from Different Angles (Orthographic Projections):
    • Looking at a 3D object from different standard positions gives different 2D views.
    • The three most common views are:
      • Front View: What you see looking directly at the front.
      • Top View: What you see looking directly down from above.
      • Side View: What you see looking directly from one side (usually specified as left or right).
    • This is crucial in engineering and design drawings to represent a 3D object completely using 2D views.
    • Example: For a simple house shape (cuboid with a triangular prism roof), the front view might show a rectangle with a triangle on top, the top view might show two rectangles (roof panels), and the side view might show a rectangle (wall) or a shape like a pentagon depending on the roof overhang.

Key Takeaways for Exams:

  • Be able to differentiate between 2D and 3D shapes.
  • Identify and count Faces, Edges, and Vertices of common polyhedrons.
  • Know and apply Euler's formula (F + V - E = 2).
  • Recognize nets for common solids (cube, cuboid, cylinder, cone, pyramid) and identify invalid nets.
  • Understand the difference between oblique and isometric sketches and their characteristics.
  • Predict the shape of cross-sections when a solid is sliced.
  • Predict the shape of shadows cast by solids under different lighting conditions.
  • Identify the Front, Top, and Side views of simple solid objects or arrangements.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed:

  1. Which of the following is a 3-Dimensional shape?
    (a) Rectangle
    (b) Circle
    (c) Triangle
    (d) Sphere

  2. A solid shape has 6 faces, 8 vertices, and 12 edges. What is this shape most likely?
    (a) Triangular Pyramid
    (b) Square Pyramid
    (c) Cube
    (d) Cone

  3. Which formula correctly relates the number of faces (F), vertices (V), and edges (E) of a polyhedron?
    (a) F + E - V = 2
    (b) F + V - E = 2
    (c) V + E - F = 2
    (d) F - V + E = 2

  4. Which of the following 2D patterns can be folded to form a cube?
    (a) A pattern with 5 squares.
    (b) A pattern with 6 squares arranged in a 2x3 rectangle.
    (c) A pattern with 6 squares forming a cross shape.
    (d) A pattern with 7 squares.

  5. A drawing of a solid shape where the front face is drawn accurately, but receding edges are drawn at an angle (often 45°) and may not be to scale, is called:
    (a) An Isometric Sketch
    (b) A Net
    (c) An Oblique Sketch
    (d) A Top View

  6. What is the shape of the cross-section obtained when a cylinder is cut vertically through its centre?
    (a) Circle
    (b) Rectangle
    (c) Triangle
    (d) Oval

  7. If you shine a torchlight directly from the top onto a cone standing on its circular base, what shape will its shadow be?
    (a) Triangle
    (b) Square
    (c) Circle
    (d) Rectangle

  8. How many vertices does a square pyramid have?
    (a) 4
    (b) 5
    (c) 6
    (d) 8

  9. Isometric sketches are typically drawn on:
    (a) Plain paper
    (b) Graph paper with squares
    (c) Isometric dot paper
    (d) Lined paper

  10. Looking directly down from above onto a solid object gives which view?
    (a) Front View
    (b) Side View
    (c) Top View
    (d) Bottom View

Answer Key:

  1. (d) Sphere
  2. (c) Cube (or Cuboid)
  3. (b) F + V - E = 2
  4. (c) A pattern with 6 squares forming a cross shape.
  5. (c) An Oblique Sketch
  6. (b) Rectangle
  7. (c) Circle
  8. (b) 5 (4 at the base, 1 apex)
  9. (c) Isometric dot paper
  10. (c) Top View

Study these notes carefully. Visualising shapes takes practice, so try drawing nets, sketches, and identifying views for different objects around you. Good luck!

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