Class 6 Mathematics Notes Chapter 5 (Chapter 5) – Exemplar Problem (English) Book

Alright class, let's get started with Chapter 5: Understanding Elementary Shapes from your NCERT Class 6 Maths Exemplar book. This chapter is fundamental for geometry, and understanding these concepts well will be very helpful for various government exams where basic mathematics and spatial reasoning are tested. Pay close attention!

Chapter 5: Understanding Elementary Shapes - Detailed Notes

1. Introduction:
Geometry is all about shapes and their properties. This chapter introduces basic geometrical ideas and ways to measure and classify shapes.

2. Measuring Line Segments:

• Line Segment: A part of a line with two fixed endpoints. It has a definite length.
• Comparison Methods:
  • Observation: Simply looking at segments to tell which is longer. Prone to errors if lengths are very close.
  • Tracing: Using tracing paper to copy one segment and compare it with another. More accurate than observation but can be cumbersome.
  • Ruler and Divider:
    • Ruler: Place the zero mark of the ruler at one endpoint and read the mark against the other endpoint. Potential Error: Incorrect positioning of the ruler or eye (parallax error).
    • Divider: Place the two points of the divider on the endpoints of the segment. Then, without changing the opening, place one point on the zero mark of the ruler and read the position of the other point. This is generally the most accurate method.

3. Angles:

• Formation: An angle is formed when two rays originate from the same endpoint (vertex). The rays are called the arms of the angle.
• Measurement Unit: Degrees (°).
• Measuring Tool: Protractor.
• Angles and Revolutions:
  • Complete Angle: One full turn. Measures 360°.
  • Straight Angle: Half a revolution (turn). Forms a straight line. Measures 180°. (Equivalent to 2 right angles).
  • Right Angle: One-quarter of a revolution (turn). Measures 90°. Often represented by a small square symbol at the vertex.
  • Acute Angle: An angle smaller than a right angle ( > 0° and < 90°).
  • Obtuse Angle: An angle larger than a right angle but smaller than a straight angle ( > 90° and < 180°).
  • Reflex Angle: An angle larger than a straight angle but less than a complete angle ( > 180° and < 360°).
• Clock Analogy: Often used to understand angles and revolutions (e.g., moving from 12 to 3 is a 90° turn or 1/4 revolution; 12 to 6 is 180° or 1/2 revolution).

4. Perpendicular Lines:

• Definition: Two lines (or line segments or rays) are said to be perpendicular if they intersect such that the angle between them is a right angle (90°).
• Notation: If line 'l' is perpendicular to line 'm', we write l ⊥ m.
• Perpendicular Bisector: A line that divides a line segment into two equal parts and is also perpendicular to it.

5. Triangles (Classification):

• A polygon with 3 sides.
• Classification by Sides:
  • Scalene Triangle: All three sides have different lengths. All three angles are usually different.
  • Isosceles Triangle: Any two sides have equal lengths. The angles opposite the equal sides are also equal.
  • Equilateral Triangle: All three sides have equal lengths. All three angles are equal (each is 60°).
• Classification by Angles:
  • Acute-angled Triangle: All three angles are acute (less than 90°).
  • Right-angled Triangle: One angle is a right angle (90°). The side opposite the right angle is called the hypotenuse.
  • Obtuse-angled Triangle: One angle is obtuse (greater than 90°). A triangle can have only one obtuse angle.

6. Quadrilaterals:

• A polygon with 4 sides.
• Types and Properties:
  • Parallelogram:
    • Opposite sides are parallel.
    • Opposite sides are equal in length.
    • Opposite angles are equal.
    • Diagonals bisect each other (cut each other into two equal parts).
  • Rectangle: (A special parallelogram)
    • All properties of a parallelogram.
    • All angles are right angles (90°).
    • Diagonals are equal in length.
  • Square: (A special rectangle and rhombus)
    • All properties of a rectangle.
    • All sides are equal in length.
    • Diagonals bisect each other at right angles (90°).
  • Rhombus: (A special parallelogram)
    • All properties of a parallelogram.
    • All sides are equal in length.
    • Diagonals bisect each other at right angles (90°). (Diagonals are not necessarily equal).
  • Trapezium (or Trapezoid):
    • Has exactly one pair of parallel opposite sides.
    • Isosceles Trapezium: Non-parallel sides are equal.
  • Kite:
    • Two distinct pairs of adjacent sides are equal in length.
    • One diagonal is the perpendicular bisector of the other.
    • One pair of opposite angles (between unequal sides) are equal.

7. Polygons:

• Definition: A simple closed figure made up entirely of line segments.
• Classification based on Number of Sides:
  • 3 sides: Triangle
  • 4 sides: Quadrilateral
  • 5 sides: Pentagon
  • 6 sides: Hexagon
  • 7 sides: Heptagon
  • 8 sides: Octagon ... and so on.
• Regular Polygon: A polygon where all sides are equal in length, and all angles are equal in measure (e.g., equilateral triangle, square).

8. Three-Dimensional Shapes (Solids):

• Shapes that have length, breadth, and height/depth.
• Key Components:
  • Faces: The flat surfaces of a solid shape.
  • Edges: The line segments where two faces meet.
  • Vertices: The points (corners) where three or more edges meet.
• Common 3D Shapes:
  • Cube: 6 square faces, 12 equal edges, 8 vertices.
  • Cuboid: 6 rectangular faces (opposite faces are identical), 12 edges (groups of 4 are equal), 8 vertices.
  • Sphere: Perfectly round, no flat faces, no edges, no vertices. Has a curved surface.
  • Cylinder: 2 circular flat faces (bases), 1 curved face, 2 circular edges, no vertices.
  • Cone: 1 circular flat face (base), 1 curved face, 1 circular edge, 1 vertex (apex).
  • Pyramid: A base (polygon) and triangular faces meeting at a common vertex (apex).
    • Triangular Pyramid (Tetrahedron): Triangular base, 3 triangular faces. (4 faces, 6 edges, 4 vertices).
    • Square Pyramid: Square base, 4 triangular faces. (5 faces, 8 edges, 5 vertices).
  • Prism: Two identical polygonal bases and rectangular side faces connecting corresponding sides of the bases.
    • Triangular Prism: 2 triangular bases, 3 rectangular faces. (5 faces, 9 edges, 6 vertices).

Key Takeaways for Exams:

• Know the definitions and properties of all shapes (2D and 3D).
• Be able to classify triangles and quadrilaterals based on sides and angles.
• Understand the different types of angles and their relation to revolutions.
• Be familiar with the terms: perpendicular lines, parallel lines, vertex, edge, face.
• Practice identifying shapes and counting their faces, edges, and vertices.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Chapter 5 concepts, suitable for practice:

1. How many right angles make a straight angle?
   (a) 1
   (b) 2
   (c) 3
   (d) 4
   Answer: (b)

2. An angle measuring 270° is an example of a:
   (a) Acute angle
   (b) Obtuse angle
   (c) Straight angle
   (d) Reflex angle
   Answer: (d)

3. A triangle with side lengths 6 cm, 6 cm, and 8 cm is called:
   (a) Scalene
   (b) Isosceles
   (c) Equilateral
   (d) Right-angled
   Answer: (b)

4. Which of the following quadrilaterals has diagonals that necessarily bisect each other at right angles?
   (a) Rectangle
   (b) Parallelogram
   (c) Rhombus
   (d) Trapezium
   Answer: (c) (Note: A square also does, but Rhombus is the more general answer among the choices that fits).

5. A polygon with 6 sides is called a:
   (a) Pentagon
   (b) Heptagon
   (c) Hexagon
   (d) Octagon
   Answer: (c)

6. How many faces does a triangular pyramid (tetrahedron) have?
   (a) 3
   (b) 4
   (c) 5
   (d) 6
   Answer: (b)

7. Which 3D shape has 1 curved face, 1 flat circular face, and 1 vertex?
   (a) Cylinder
   (b) Sphere
   (c) Cone
   (d) Cube
   Answer: (c)

8. If two lines intersect and form four equal angles, the lines are:
   (a) Parallel
   (b) Perpendicular
   (c) Intersecting but not perpendicular
   (d) Skew
   Answer: (b) (Each angle would be 360°/4 = 90°)

9. A triangle has angles measuring 50°, 90°, and 40°. What type of triangle is it based on its angles?
   (a) Acute-angled
   (b) Obtuse-angled
   (c) Right-angled
   (d) Equilateral
   Answer: (c)

10. Which statement is true for a square?
    (a) Only opposite sides are equal.
    (b) Diagonals are unequal.
    (c) All angles are acute.
    (d) All sides are equal and all angles are 90°.
    Answer: (d)

Study these notes thoroughly and practice identifying and classifying shapes. Good luck with your preparation!