Topics
Number Systems
Number Systems
Algebra
Polynomials
Linear Equations in Two Variables
Algebraic Expressions
Algebraic Identities
Coordinate Geometry
Geometry
Introduction to Euclid’S Geometry
Lines and Angles
Triangles
Quadrilaterals
- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- Angle Sum Property of a Quadrilateral
- Types of Quadrilaterals
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Another Condition for a Quadrilateral to Be a Parallelogram
- The Mid-point Theorem
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
Area
Circles
- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- Angle Subtended by a Chord at a Point
- Perpendicular from the Centre to a Chord
- Circles Passing Through One, Two, Three Points
- Equal Chords and Their Distances from the Centre
- Angle Subtended by an Arc of a Circle
- Cyclic Quadrilateral
Constructions
Mensuration
Areas - Heron’S Formula
Surface Areas and Volumes
Statistics and Probability
Statistics
Probability
definition
Area: The amount of surface enclosed by a closed figure is called its area.
notes
Area:
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The amount of surface enclosed by a closed figure is called its area.
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A polygon is a plane shape with straight sides. The area of a polygon measures the size of the region enclosed by the polygon.
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It is measured in units squared.
Visualisation of covered Area on the graph:
It is difficult to tell the area of the figure just by looking at the figure.
Place them on a squared paper or graph paper where every square measure 1 cm × 1 cm.
Make an outline of the figure.
Look at the squares enclosed by the figure. Some of them are completely enclosed, some half, some less than half and some more than half.
The area is the number of centimeter squares that are needed to cover it
But there is a small problem: the squares do not always fit exactly into the area you measure. We get over this difficulty by adopting a convention:
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The area of one full square is taken as 1 sq unit. If it is a centimeter
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square sheet, then the area of one full square will be 1 sq cm.
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Ignore portions of the area that are less than half a square.
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If more than half of a square is in a region, just count it as one square.
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If exactly half the square is counted, take its area as `1/2`sq unit.
Such a convention gives a fair estimate of the desired area.
| Covered area | Number | Area estimate (sq. units) |
| (i) Fully-filled squares | 101 | 101 |
| (ii) Half-filled squares | 4 | 4 × `1/2` |
| (iii) More than half-filled squares | 21 | 21 |
| (iv) Less than half-filled squares | 23 | 0 |
Total area = 101 + 4 × `1/2` + 21 = 126`1/2` sq. Units.
Example
By counting squares, estimate the area of the figure.
| Covered area | Number |
Area estimate |
| (i) Fully-filled squares | 11 | 11 |
| (ii) Half-filled squares | 3 | `3 xx 1/2` |
| (iii) More than half-filled squares | 7 | 7 |
| (iv) Less than half-filled squares | 5 | 0 |
Total area = 11 + 3 × `1/2 + 7 = 19 1/2` sq units.
Example
By counting squares, estimate the area of the figure.
| Covered area | Number | Area estimate (sq units) |
| (i) Fully-filled squares | 1 | 1 |
| (ii) Half-filled squares | - | - |
| (iii) More than half-filled squares | 7 | 7 |
| (iv) Less than half-filled squares | 9 | 0 |
Total area = 1 + 7 = 8 sq units.
