24 .AREA
FUNDEMENTAL CONCEPTS
I.RESULTS ON TRIANGLES:
1.Sum of
the angles of a triangle is 180 degrees.
2.Sum of any two sides of a triangle is greater than the third side.
3.Pythagoras theorem:
In a right angle triangle,
(Hypotenuse)^2 = (base)^2 + (Height)^2
4.The line joining the midpoint of a side of a triangle to the opposite vertex is called the
MEDIAN
5.The point where the three medians of a triangle meet is called
CENTROID
.
Centroid divides each of the medians in the ratio 2:1.
6.In an isosceles triangle, the altitude from the vertex bisects the base
7.The median of a triangle divides it into two triangles of the same area.
8.Area of a triangle formed by joining the midpoints of the sides of a given triangle is
onefourth of the area of the given triangle.
II.RESULTS ON QUADRILATERALS:
1.
The diagonals of a parallelogram bisects each other .
2.
Each diagonal of a parallelogram divides it into two triangles of the same area
3.
The diagonals of a rectangle are equal and bisect each other.
4.
The diagonals of a square are equal and bisect each other at right angles.
5.
The
diagonals of a rhombus are unequal and bisect each other at right angles.
6.
A parallelogram and a rectangle on the same base and between the same parallels
are equal in area.
7.
Of all the parallelograms of a given sides , the parallelogram which is a rectangle
has the greatest area.
IMPORTANT FORMULAE
I
.1.Area of a rectangle=(length*breadth)
Therefore length = (area/breadth) and breadth=(area/length)
2.Perimeter of a rectangle = 2*(length+breadth)
II
.Area of a square = (side)^2 =1/2(diagonal)^2
III
Area of four walls of a room = 2*(length + breadth)*(height)
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1.Area of the triangle=1/2(base*height)
2. Area of a triangle = (s*(sa)(sb)(sc))^(1/2), where a,b,c are the sides of a triangle
and s= ½(a+b+c)
3.Area of the equilateral triangle =((3^1/2)/4)*(side)^2
4.Radius of incircle of an equilateral triangle
of side a=a/2(3^1/2)
5.Radius of circumcircle of an equilateral triangle of side a=a/(3^1/2)
6.Radius of incircle of a triangle of area del and semiperimeter S=del/S
V.
1.Area of the parellogram =(base *height)
2.Area of the rhombus=1/2(product of the diagonals)
3.Area of the trapezium=1/2(size of parallel sides)*distance between them
VI
1.Area of a circle =pi*r^2,where r is the radius
2. Circumference of a circle = 2∏R.
3. Length of an arc = 2∏Rθ/(360) where θ is the central angle
4. Area of a sector = (1/2) (arc x R) = pi*R^2*θ/360.
VII
. 1. Area of a semicircle = (pi)*R^2.
2. Circumference of a semicircle = (pi)*R.
SOLVED EXAMPLES
Ex.1. One side of a rectangular field is 15 m and one of its diagonals is 17 m. Find
the area of the field.
Sol
. Other side = ((17)
2
 (15)
2)(1/2
) = (289 225)
(1/2)
= (64)
(1/2)
= 8 m.
Area = (15 x 8) m
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 Spring '11
 vinh

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