AREA
FUNDAMENTAL
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 bi-sects 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 one-fourth 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)
IV 1.Area of the
triangle=1/2(base*height)
2. Area of a triangle =
(s*(s-a)(s-b)(s-c))^(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 semi-circle = (pi)*R^2.
2. Circumference of a semi-circle =
(pi)*R.
VOLUME
AND SURFACE AREA
FORMULAE
I.
CUBOID
Let length = 1, breadth =
b and height = h units. Then, 1. Volume = (1 x b x h) cubic units.
2. Surface area= 2(lb + bh + lh) sq.units.
3. Diagonal.=Öl2 +b2 +h2 units
II.
CUBE
Let each edge of a cube
be of length a. Then,
1. Volume = a3 cubic units.
2. Surface area = 6a2 sq. units.
3. Diagonal = Ö3 a units.
III.
CYLINDER
Let radius of base = r
and Height (or length) = h. Then,
1. Volume = (P r2h) cubic units.
2. Curved surface area = (2P rh). units.
3. Total surface area =2Pr (h+r) sq. units
IV.
CONE
Let radius of base = r
and Height = h. Then,
1. Slant height, l =Ö h2+r2
2.
Volume = (1/3) Pr2h cubic units.
3. Curved surface area = (Prl) sq. units.
4. Total surface area = (Prl + Pr2 ) sq. units.
V. SPHERE
Let the radius of the
sphere be r. Then,
1. Volume = (4/3)Pr3 cubic units.
2. Surface area = (4Pr2) sq. units.
VI.
HEMISPHERE
Let the radius of a
hemisphere be r. Then,
1. Volume = (2/3)Pr3 cubic units.
2. Curved surface area = (2Pr2) sq. units.
3. Total surface area = (3Pr2) units.
Remember: 1 litre = 1000 cm3.
OR 