Class 6 Mathematics Notes Chapter 5 (Understanding Elementary Shapes) – Mathematics Book
Detailed Notes with MCQs of Chapter 5, 'Understanding Elementary Shapes'. This chapter builds the foundation for geometry, so understanding these concepts clearly is crucial, not just for your class exams but also for future competitive exams where geometry often features. We'll break it down systematically.
Chapter 5: Understanding Elementary Shapes - Detailed Notes
1. Introduction:
This chapter introduces basic geometric shapes and ways to measure and compare them. We learn about line segments, angles, different types of polygons (like triangles and quadrilaterals), and basic 3D shapes.
2. Measuring Line Segments:
- Line Segment: A fixed portion of a line with two endpoints. It has a definite length.
- Comparison Methods:
- Comparison by Observation: Just by looking, we can sometimes tell if one segment is longer than another. This is not accurate and prone to errors.
- Comparison by Tracing: Trace one segment onto paper and place it over the other. Better than observation, but still not precise.
- Comparison using Ruler and Divider:
- Ruler: Place the zero mark of the ruler at one endpoint and read the mark against the other endpoint. Potential for error due to thickness of the ruler markings and positioning of the eye (parallax error).
- Divider: Place the endpoints of the divider on the endpoints of the segment. Then, without disturbing the opening, place one end of the divider on the zero mark of the ruler and read the measurement where the other end falls. This is generally the most accurate method.
3. Angles - Right and Straight:
- Angle: Formed when two rays (or lines/line segments) meet at a common endpoint (vertex). Measured in degrees (°).
- Directions and Turns: Turning is associated with angles.
- North, South, East, West are the main directions.
- Turning from North clockwise to East is a Right Angle (90°).
- Turning from North clockwise to South is a Straight Angle (180°). This is equivalent to two right angles.
- Turning from North clockwise back to North is a Complete Angle (360°). This is equivalent to four right angles.
- Revolution: A complete turn (360°) is one revolution.
- Right Angle = 1/4 Revolution
- Straight Angle = 1/2 Revolution
- Complete Angle = 1 Revolution
- Clock Angles: The hands of a clock move and form angles.
- From 12 to 3: 1/4 revolution or 90°.
- From 12 to 6: 1/2 revolution or 180°.
- From 12 to 9: 3/4 revolution or 270°.
- From 12 back to 12: 1 revolution or 360°.
- Each hour mark represents 360°/12 = 30°.
4. Angles - Acute, Obtuse, and Reflex:
- Types of Angles based on measure:
- Acute Angle: Measure is greater than 0° but less than 90°. ( < 90° )
- Right Angle: Measure is exactly 90°. ( = 90° )
- Obtuse Angle: Measure is greater than 90° but less than 180°. ( > 90° and < 180° )
- Straight Angle: Measure is exactly 180°. ( = 180° )
- Reflex Angle: Measure is greater than 180° but less than 360°. ( > 180° and < 360° )
- Complete Angle: Measure is exactly 360°. ( = 360° )
- Measuring Angles: We use a Protractor to measure angles in degrees. Place the midpoint of the protractor on the vertex of the angle and align the base line with one arm of the angle. Read the measure where the other arm crosses the scale.
5. Perpendicular Lines:
- When two lines intersect and the angle between them is a right angle (90°), the lines are said to be perpendicular to each other.
- Symbol: ⊥ (e.g., Line AB ⊥ Line CD)
- Perpendicular Bisector: A line that is perpendicular to a given line segment and divides it into two equal parts.
6. Classification of Triangles:
- Triangle: A polygon with 3 sides and 3 angles.
- Classification based on Sides:
- Scalene Triangle: All three sides have different lengths. All three angles are also 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 based on Angles:
- Acute-angled Triangle: All three angles are acute (less than 90°).
- Right-angled Triangle: One angle is a right angle (exactly 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.
7. Quadrilaterals:
- Quadrilateral: A polygon with 4 sides and 4 angles.
- Types of Quadrilaterals (based on properties):
- Rectangle: A parallelogram with all angles equal to 90°. Opposite sides are equal in length. Diagonals are equal and bisect each other.
- Square: A rectangle with all four sides equal. It is also a rhombus. All angles are 90°. Diagonals are equal, bisect each other at right angles (90°).
- Parallelogram: A quadrilateral with opposite sides parallel and equal in length. Opposite angles are equal. Diagonals bisect each other.
- Rhombus: A parallelogram with all four sides equal in length. Opposite angles are equal. Diagonals bisect each other at right angles (90°).
- Trapezium (or Trapezoid): A quadrilateral with at least one pair of opposite sides parallel.
8. Polygons:
- Polygon: 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).
- Irregular Polygon: A polygon that is not regular (sides or angles or both are unequal).
9. Three-Dimensional Shapes (Solids):
- Shapes that have length, breadth, and height (or depth). They occupy space.
- 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. (e.g., Dice)
- Cuboid: 6 rectangular faces (opposite faces are identical), 12 edges (edges meeting at a vertex can have different lengths), 8 vertices. (e.g., Matchbox, Book)
- Sphere: No flat faces, no edges, no vertices. Perfectly round. (e.g., Ball)
- Cylinder: 2 circular flat faces, 1 curved surface, 2 circular edges, no vertices. (e.g., Can, Pipe)
- Cone: 1 circular flat face (base), 1 curved surface, 1 circular edge, 1 vertex (apex). (e.g., Ice cream cone, Birthday cap)
- Pyramid: Has a polygon base and triangular faces that meet at a common vertex (apex).
- Square Pyramid: Base is a square, 4 triangular faces, 8 edges, 5 vertices.
- Triangular Pyramid (Tetrahedron): Base is a triangle, 3 triangular faces, 6 edges, 4 vertices. (All faces are triangles).
Multiple Choice Questions (MCQs):
-
How many degrees are there in a straight angle?
(a) 90°
(b) 180°
(c) 270°
(d) 360° -
An angle whose measure is greater than 90° but less than 180° is called:
(a) Acute angle
(b) Right angle
(c) Obtuse angle
(d) Reflex angle -
If you stand facing North and turn clockwise to face South, what fraction of a revolution have you turned?
(a) 1/4
(b) 1/2
(c) 3/4
(d) 1 -
A triangle with all three sides of different lengths is called:
(a) Equilateral triangle
(b) Isosceles triangle
(c) Scalene triangle
(d) Right-angled triangle -
Which of the following quadrilaterals has all sides equal and all angles equal to 90°?
(a) Rectangle
(b) Rhombus
(c) Square
(d) Parallelogram -
How many faces does a cuboid have?
(a) 4
(b) 6
(c) 8
(d) 12 -
A polygon with 6 sides is called a:
(a) Pentagon
(b) Hexagon
(c) Heptagon
(d) Octagon -
What type of angle is formed by the hands of a clock at 3 o'clock?
(a) Acute angle
(b) Right angle
(c) Obtuse angle
(d) Straight angle -
Which instrument is generally considered most accurate for comparing the lengths of two line segments?
(a) Ruler
(b) Protractor
(c) Divider
(d) Tracing paper -
A triangular pyramid (Tetrahedron) has:
(a) 4 faces, 6 edges, 4 vertices
(b) 5 faces, 8 edges, 5 vertices
(c) 3 faces, 6 edges, 4 vertices
(d) 4 faces, 4 edges, 4 vertices
Answer Key for MCQs:
- (b) 180°
- (c) Obtuse angle
- (b) 1/2
- (c) Scalene triangle
- (c) Square
- (b) 6
- (b) Hexagon
- (b) Right angle
- (c) Divider
- (a) 4 faces, 6 edges, 4 vertices
Remember to revise these concepts thoroughly. Understanding the definitions and properties is key to solving problems based on shapes. Good luck with your preparation!