1. welcome

2. P.Dr Vitthalrav Vikhe Patil Vidyalaya,Loni
Name- -Nimbore v p

3. GEOMETRY
MENSURATION

4. Let us understand the concept of area
Observe the adjoining figures. Do given two figures represent same set of points? Here fig.9.1 represent a region which includes a triangle and its interior. While fig.9.2 represent only a triangle. The union of a triangle and its Interior is called triangular region.. IT has definite area,whereas Triangle has no area. So just like triangular region,there Are rectangular, circular and polygonal Region etc. In this chapter we are referring area of Triangular, circular and polygonal region etc.as area of triangle,area of circle and area of polygon etc.

5. Area of rectangular region
In fig.9.3 square APQR is a square whose side is1cm so its area is 1sq. cm. which is used for measuring area.There are 24 such squares in rectangular region ABCD.So its total area is 24 sq.cm. But length of rectangle ABCD, AB=6 cm and its Breadth AD=4cm. So the product of length and breadth is 6 cm×4cm=24 sq.cm. Area of any region may be in sq.cm (cm2),sq.m(m2) Sq,km (km2) or sq units ,etc. D C R Q
1cm A P B
1cm

6. Area of triangle
When triangle with common base or congruent bases lie between two parallel lines, their areas are equal. When a triangle and a parallelogram are on the common base or congruent bases and between the same parallel lines ,then the area of the triangle is half the area of the parallelogram

7. Area and Perimeter of a quadrilateral
Ex. The length of a rectangular plot of land is twice its breadth. If the perimeter of the plot is 420m .find its area
Solution: Let the breadth of a rectangle be x metres . .Its length =2 x metres
Perimeter of rectangle = 2(l+b)
2(l+b)=420
2(2x+x)=420
2×3 x=420
x=420/6
X=70
Breadth =700m and Length=2x=2×70=140m
Area of rectangular polt=l×b=140×70=9800
Area of rectangular polt is 9800sq.m. .

8. Area and Perimeter of the circle and semicircle
We have learnt the concept of area. Also we know perimeter of circle means its circumference. Ratio of circumference of a circle to its diameter is constant. It is denoted by greek Letter pi Its an irrational number.We take its approximate value as 22/7. The ratio of circumference of any circle to its diameter is constant, can be proved in the following way. Suppose there are two concentric circles with centre P Let their radii be r1 and r2 as shown in fig.9.13 If we draw n-regular polygons inscribed in each circle so that each side of regular polygon inscribed in larger circle is seg CD and in smaller circle is seg AB

9. Now PCD is an isosceles triangle(PC=PD=r2) Also PAB is an isosceles triangle (PA=PB=r1) M angle P=360/n (where number of sides of polygon is n) In PCD and PAB Angle APB congurent angle CPD……..(common angle) PC/PA=PD/PB=r2/r1 …….....(i) PCD PAB(by SAS test) CD/AB=r2/r1 (C.S.T.P) AB/r1=CD/r2 n×AB/r1= n×CD/r2 (multiplying both the sides by n ) If P1 and p2 are the perimeters of regular polygons respectively. n×AB=P1 and n×CD=P2 P1/r1=P2/r2 Suppose the circumference of regular polygons respectively. If n is very large then perimeter will be the circumference. P1→C1 and P2→C2means p1 will be c1 and p2 will be c2 C1/r1=c2/r2 It means c1/2r1=c2/2r2 C1/D1=C2/D2 (2r1=d1 and 2r2=d2) The ratio of circumference of the circle and its diameter is always constant. This constant is denote by greek letter pi For any circle C/2R=Pi i.e. C=2 PI r or C= pi d where d is the diameter Perimeter of a semicircle =1/2×Cirumference+ diameter =1/2 c +d =pi r +d =1/2 pi r2

10. Area of regular polygon
Definition: A simple closed figure formed by line segments is called a polygon.
Regular polygon: A polygon in which all sides and all angle are congruent is called regular polygon
Centre of a polygon :The inscribed and circumscribed circles of a regular polygon have the same centre this central point of a polygon
In-radius :The length of the perpendicular form the central point of a angular polygon to any of its side is called the radius of the inscribed circle of that polygon
Circum- radius :The line segment joining central point of a polygon. To any vertex is called the radius of circumscribed circle of the polygon .it is circum-radius .it is denoted by R

11. Area of a regular polygon of n-sides
Area of a regular polygon of n sides=1/2 nar(hera,n=numbers of sides,a=length of the side and r is the radius of the inscribed circle)
Perimeter of n-sided regular polygon =na form(1),the perimeter of n-sided regular polygon =2a/r

12. Particular Cases
Regular hexagon : Area of a regular hexagon with side of the length a=3√3/2a2
Regular octagon :Area of a regular octagon with side of the length a=2(1+√2)a2.