AP · Class 8 · 🧮 Maths · Chapter 15
ch15
ప్రజాస్వామ్యంఎన్నికలుప్రభుత్వ జవాబుదారీతనంప్రాతినిధ్య ప్రజాస్వామ్యంనియంతృత్వం
ఈ అధ్యాయం ప్రజాస్వామ్య ప్రాతినిధ్యం, ప్రభుత్వ జవాబుదారీతనం మరియు ఎన్నికల ప్రాముఖ్యతను వివరిస్తుంది. విద్యార్థులు నియంతృత్వం, ప్రజాస్వామ్యం, ప్రత్యక్ష మరియు పరోక్ష ప్రజాస్వామ్యం మధ్య తేడాలను అర్థం చేసుకోవడంపై దృష్టి సారిస్తుంది. ప్రభుత్వాలు ప్రజల సంక్షేమం కోసం పని చేయడానికి ఎన్నికలు ఎలా ప్రధాన ప్రోత్సాహాన్ని అందిస్తాయో ఈ అధ్యాయం వివరిస్తుంది. ఇది ప్రజాస్వామ్యంలో తనిఖీలు మరియు బ్యాలెన్స్ల ప్రాథమిక భావనను అర్థం చేసుకోవడానికి సహాయపడుతుంది.
Introduction to 2D Shapes: Polygons
2D Shapes: Polygons
- Definition: A polygon is a closed figure made up of three or more line segments. These segments are called sides.
- Classification by Number of Sides:
- 3 sides: Triangle
- 4 sides: Quadrilateral
- 5 sides: Pentagon
- 6 sides: Hexagon
- 7 sides: Heptagon
- 8 sides: Octagon
- 9 sides: Nonagon
- 10 sides: Decagon
- Types of Polygons:
- Regular Polygon: All sides are equal in length and all interior angles are equal. E.g., equilateral triangle, square.
- Irregular Polygon: Sides and/or angles are not all equal.
- Convex Polygon: All interior angles are less than 180°. All diagonals lie entirely inside the polygon.
- Concave Polygon: At least one interior angle is greater than 180°. At least one diagonal lies partially or entirely outside the polygon.
- Angle Sum Property of Polygons:
- Sum of interior angles of an n-sided polygon = \((n-2) \times 180°\).
- Each interior angle of a regular n-sided polygon = \(\frac{(n-2) \times 180°}{n}\).
- Sum of exterior angles of any convex polygon = 360°.
- Each exterior angle of a regular n-sided polygon = \(\frac{360°}{n}\).
- Relationship: Interior angle + Exterior angle = 180°.
Special Quadrilaterals
- Parallelogram: Opposite sides are parallel and equal. Opposite angles are equal. Diagonals bisect each other.
- Rectangle: A parallelogram with all angles 90°. Diagonals are equal.
- Rhombus: A parallelogram with all sides equal. Diagonals bisect each other at 90°.
- Square: A rectangle with all sides equal (or a rhombus with all angles 90°). Diagonals are equal and bisect each other at 90°.
- Trapezium (Trapezoid): One pair of opposite sides is parallel.
- Isosceles Trapezium: Non-parallel sides are equal. Base angles are equal.
- Kite: Two pairs of adjacent sides are equal. Diagonals are perpendicular. One diagonal bisects the other.
గుర్తుంచుకోండి
The sum of the interior angles of a triangle is always 180°. This is the base for the polygon angle sum formula.
తప్పుడు అభిప్రాయం
Don't confuse convex and concave polygons. A simple test: if any part of a diagonal goes outside the shape, it's concave.
Perimeter and Area of 2D Shapes
Perimeter and Area of 2D Shapes
- Perimeter: The total length of the boundary of a closed figure.
- Area: The measure of the surface enclosed by a closed figure.
Formulas for Perimeter and Area
- Square:
- Perimeter = \(4 \times \text{side}\)
- Area = \(\text{side} \times \text{side} = \text{side}^2\)
- Rectangle:
- Perimeter = \(2 \times (\text{length} + \text{breadth})\)
- Area = \(\text{length} \times \text{breadth}\)
- Triangle:
- Perimeter = Sum of all three sides
- Area = \(\frac{1}{2} \times \text{base} \times \text{height}\)
- Parallelogram:
- Perimeter = \(2 \times (\text{adjacent side 1} + \text{adjacent side 2})\)
- Area = \(\text{base} \times \text{height}\)
- Rhombus:
- Perimeter = \(4 \times \text{side}\)
- Area = \(\frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2\)
- Trapezium:
- Perimeter = Sum of all four sides
- Area = \(\frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height}\)
- Circle:
- Circumference (Perimeter) = \(2 \pi r\) or \(\pi d\) (where r = radius, d = diameter)
- Area = \(\pi r^2\)
Units of Measurement
- Perimeter is measured in linear units (cm, m, km).
- Area is measured in square units (cm², m², km²).
- Conversions:
- 1 m = 100 cm
- 1 km = 1000 m
- 1 m² = 100 cm \(\times\) 100 cm = 10000 cm²
- 1 km² = 1000 m \(\times\) 1000 m = 1,000,000 m²
- 1 hectare = 10000 m²
- 1 acre = approx. 4047 m²
సూత్రం
Area of a Trapezium: \(A = \frac{1}{2} (a+b)h\), where 'a' and 'b' are the lengths of the parallel sides and 'h' is the perpendicular distance between them.
సూచన
Always pay attention to the units given in the problem and ensure your final answer is in the correct units. Convert all measurements to a consistent unit before calculation.
Introduction to 3D Shapes: Polyhedrons
3D Shapes: Polyhedrons
- Definition: A 3D shape that has flat polygonal faces, straight edges, and sharp corners or vertices.
- Key Terms:
- Face: A flat surface of a 3D shape (a polygon).
- Edge: A line segment where two faces meet.
- Vertex: A point where three or more edges meet.
- Euler's Formula for Polyhedrons: For any polyhedron, the number of faces (F), vertices (V), and edges (E) are related by the formula: F + V - E = 2.
Types of 3D Shapes
- Prism: A polyhedron with two identical and parallel polygonal bases, and rectangular (or parallelogram) lateral faces.
- Triangular Prism: Bases are triangles.
- Rectangular Prism (Cuboid): Bases are rectangles.
- Cube: A special rectangular prism where all faces are squares.
- Pyramid: A polyhedron with a polygonal base and triangular lateral faces that meet at a common vertex (apex).
- Triangular Pyramid (Tetrahedron): Base is a triangle.
- Square Pyramid: Base is a square.
- Non-Polyhedrons: Shapes with curved surfaces.
- Cylinder: Two circular bases, one curved surface.
- Cone: One circular base, one curved surface meeting at an apex.
- Sphere: A perfectly round 3D object, all points on its surface are equidistant from its center.
ముఖ్యమైనది
Euler's Formula (F + V - E = 2) is a fundamental property of all polyhedrons. It's often tested in exams.
గుర్తుంచుకోండి
A cube is a special case of a cuboid, and a square pyramid is a type of pyramid. Understand the hierarchy.
Surface Area and Volume of 3D Shapes
Surface Area and Volume of 3D Shapes
- Surface Area: The total area of all the faces (or surfaces) of a 3D object.
- Lateral Surface Area (LSA) / Curved Surface Area (CSA): Area of only the side faces, excluding bases.
- Total Surface Area (TSA): Area of all faces, including bases.
- Volume: The amount of space occupied by a 3D object.
Formulas for Surface Area and Volume
- Cuboid (length 'l', breadth 'b', height 'h'):
- LSA = \(2h(l+b)\)
- TSA = \(2(lb + bh + hl)\)
- Volume = \(l \times b \times h\)
- Cube (side 'a'):
- LSA = \(4a^2\)
- TSA = \(6a^2\)
- Volume = \(a^3\)
- Cylinder (radius 'r', height 'h'):
- CSA = \(2 \pi r h\)
- TSA = \(2 \pi r h + 2 \pi r^2 = 2 \pi r (h+r)\)
- Volume = \(\pi r^2 h\)
- Cone (radius 'r', height 'h', slant height 'l'):
- \(l = \sqrt{r^2 + h^2}\)
- CSA = \(\pi r l\)
- TSA = \(\pi r l + \pi r^2 = \pi r (l+r)\)
- Volume = \(\frac{1}{3} \pi r^2 h\)
- Sphere (radius 'r'):
- Surface Area = \(4 \pi r^2\)
- Volume = \(\frac{4}{3} \pi r^3\)
- Hemisphere (radius 'r'):
- CSA = \(2 \pi r^2\)
- TSA = \(3 \pi r^2\) (includes the circular base)
- Volume = \(\frac{2}{3} \pi r^3\)
Units of Measurement
- Surface Area is measured in square units (cm², m²).
- Volume is measured in cubic units (cm³, m³).
- Capacity: Volume of liquid a container can hold. Often measured in litres.
- 1 litre = 1000 cm³
- 1 m³ = 1000 litres = 1 kilolitre
సూత్రం
Remember the difference between LSA/CSA and TSA. LSA excludes the area of the bases, while TSA includes it.
సూచన
For problems involving melting and recasting of solids, the volume remains constant. For problems involving painting or covering, it's about surface area.
Visualising 3D Shapes
Visualising 3D Shapes
- Views of 3D Objects: We can represent 3D objects in 2D by drawing their different views.
- Front View: What you see when looking directly at the front of the object.
- Side View: What you see when looking directly from the side of the object.
- Top View: What you see when looking directly down on the object.
- Nets for 3D Shapes: A net is a 2D pattern that can be folded to form a 3D shape. Understanding nets helps in visualising the faces and calculating surface area.
- Cube Net: Six squares arranged in various patterns (e.g., a 'T' shape).
- Cuboid Net: Six rectangles.
- Cylinder Net: Two circles and one rectangle.
- Cone Net: One circle and one sector of a circle.
- Pyramid Net: A polygon base and triangular faces.
- Mapping Space Around Us: Using concepts like coordinates to locate objects in 3D space (though more advanced in higher grades, basic understanding of position is key).
సూచన
Practise drawing different views (front, side, top) of common objects. This improves spatial reasoning. Also, be able to identify a 3D shape from its net.
