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
 Another Condition for a Quadrilateral to Be a Parallelogram
 Theorem of Midpoints of Two Sides of a Triangle
 Property: The Opposite Sides of a Parallelogram Are of Equal Length.
 Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
 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:

The amount of surface enclosed by a closed figure is called its area.

A polygon is a plane shape with straight sides. The area of a polygon measures the size of the region enclosed by the polygon.

It is measured in units squared.
Area of Regular Shapes
Area of shapes such as circle, triangle, square, rectangle, parallelogram, etc. are the region occupied by them in space. Polygon shape: A polygon is a twodimensional shape that is formed by straight lines. The examples of polygons are triangles, hexagons and pentagons.
Area of Square
Area of Rectangle
Area of Triangle
Area of Irregular Shapes:
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:

The area of one full square is taken as 1 sq unit. If it is a centimeter

square sheet, then the area of one full square will be 1 sq cm.

Ignore portions of the area that are less than half a square.

If more than half of a square is in a region, just count it as one square.

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) Fullyfilled squares  101  101 
(ii) Halffilled squares  4  4 × `1/2` 
(iii) More than halffilled squares  21  21 
(iv) Less than halffilled 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) Fullyfilled squares  11  11 
(ii) Halffilled squares  3  `3 xx 1/2` 
(iii) More than halffilled squares  7  7 
(iv) Less than halffilled 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) Fullyfilled squares  1  1 
(ii) Halffilled squares     
(iii) More than halffilled squares  7  7 
(iv) Less than halffilled squares  9  0 
Total area = 1 + 7 = 8 sq units.