formula class IX surface areas and volume
CLASS-IX
SURFACE AREAS AND VOLUMES
Area of Plane Figures
Some Definitions:
1.Perimeter: Perimeter of a plane figure is the measure of
the length of its boundary. Perimeter of planes ⬠
= AB + BC + CD + DE + EA.
2.Area: Area of a plane figure is the measure of the region
enclosed by it. Area of plane figure (inner region)= shaded region
ABCDE
Perimeter & Area of Different Plane figures:
1.Triangle: Let ⊿ABC has sides a, b, c.
(i)the perimeter of a triangle is = a + b + c
generally it is denoted by 2s.
then 2s = (a +b+c).
and s=1/2 (a+b+c)
s is called semi-perimeter of triangle.
(ii)Area of⊿ ABC = 1/2 Base x height
Area of ⊿ABC=1/2 BC × AD
Area of ⊿ in terms of semi-perimeter
= √s(s - a)(s-b)(s-c)
This called Heron's formula:
2.Right Angled Triangle: ∠B is right angle.
then (i) Perimeter = a + b + c
(ii) Area =1/2× a ×b
3.Equilateral triangle: Sides of Equilateral ⊿ABC are a.
(i)Perimeter = 3a
(ii) Area =√3/4 a2
4.Rectangle: Rectangle ABCD has its length l and breadth b, then
(i) Perimeter=2 (1+ b)
(ii) Area=1×b
5.Square: Square ABCD has sides a unit then
(i)Perimeter =4 a
(ii) Area=a2
6.Parallelogram: For a parallelogram whose adjacent sides
are a and b we have
(i) Perimeter = 2(a + b)
(ii) Area= Base x height = 2(Area of ⊿ ABC)
7.Rhombus: If d1 & d2 are the diagonals of rhombus ABCD then
(i)Perimeter = 4 x side =2.√d(diagonal 1 ) 2(square ) + d(diagonal 2) 2square
(ii) Area= 1/2 d1 × d2
and the sides of rhombus are-
√d1(diagonal one ) 2 (square) + d2(diagonal 2 square ) / 2 each
8.Trapezium: For a trapezium || sides are a & b units and
length between them is h units.
Area= 1/2 h (a+ b)
To find the area of a Quadrilateral, convert it into two ⊿ 's.
Polygons
A polygon is a plane figure enclosed by the line segments which are known as the sides of polygon.
Regular polygon
If a polygon has all the sides of same length then it is known as a regular polygon.
Internal Angles of a regular polygon
Each internal angle of a regular polygon of n sides is equal to (n-2 /n x180°)
Parallelopiped
A solid bounded by three pairs of parallel plane surfaces is called a parallelopiped. The plane surfaces are known as the faces of the parallelopiped.
Cuboid
A parallelopiped whose faces are rectangles and adjacent faces
are perpendicular is called a rectangular parallelopiped or a cuboid.
In a cuboid
1. AB = EF = CD =HG
2.AE =BF= DH= CG
3.AD = BC = EH=FG
first set of sides are called length of cuboid. Second set of sides
are called height of cuboid and third set of sides are called breadth
of cuboid.
Volume of cuboid= length x breadth x height = lbh
Lateral surface area = 2 height (length + breadth) = 2 h (l+b)
Total surface area=2(length × height +height x breadth + breadth x length)
Area of four walls = Lateral surface area = 2(lb + bh + hl)
Diagonal of the cuboid = √12 + b2 + h2
Cube
If the length of each edge of a cuboid is same then, it is called cube.
Let the length of edge is 'a'.
Volume of cube=a3
Lateral surface area = 4 a2
Total surface area = 6 a2
Diagonal of the cube = √3 a
Right Circular cylinder
A right circular cylinder is a solid generated by the revolution of a
rectangle about one of its side. Let h be height and r be the radius of its base.
volume of cylinder=πr2h
lateral/curved surface area= 2π rh
total surface area= 2π r(h+r)
Hollow cylinder
A solid bounded by two coaxial cylinders of the same height and
different radii is called a hollow cylinder.
Volume of Hollow cylinder = π h (R2-r2)
Lateral/curved surface area = 2 πh (R+r)
Total surface area = 2π (R +r) (h+R-r)
Right circular cone
A right circular cone is a solid generated by revolving a right angled triangle about its base.
Height of cone: The length segment CO is called height of cone and denoted by 'h'.
Slant height of cone: The length of segment AC (or BC) is called
slant height of cone. and denoted by 'l'.
Radius of cone: The radius of the base (circle) is called radius
of cone and denoted by 'r'.
So it is evident from figure
l=√h2+r2
Volume of cone=1/3 πr2h
Lateral/Curved surface Area =πrl
Total surface Area =πr (l+r)
Sphere
A sphere can be described as the set-of points in space which are
equidistant from a fixed point.
Let the radius of sphere be r
Volume of sphere= 4/3πr3
Surface area of sphere=4 πr2
Note: In case of sphere, lateral surface area = total surface area
Hemisphere
Let the radius of hemisphere be r.
Volume of hemisphere= 2/3 πr3
Lateral/curved surface area = 2πr2
Total surface area= 3πr2
Spherical shell
The difference of two solid concentric spheres is called a spherical
shell. A spherical shell has finite thickness, which is the difference
of the radii of the two solid spheres.
Volume of spherical shell= 4/3π(R3-r3)
Surface Area of spherical shell = 4π(R2+r2)
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Name Figure Lateral/curved Total surface area Volume Nomenclature
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1.cuboid b📖 2(1+b)xh 2(l b+b h+h l) 1xbxh l=length b= breadth
h= height
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2.Diagonal of cuboid = √l2+b2+h2
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3.Cube 🎁 41 2 61 2 l 3 l= edge of cube
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4.Diagonal of cube= l√3
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5.Right circular cylinder 2πrh 2πr(r+h) πr2h r= radius
🔋 h=height
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6.Right circular cone πrl πr(l+r) 1/3 πr2h r=radius of 🍧 base
h=height
l=slant height
=√ r2+h2
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7.Sphere 😆 4πr2 4πr2 4/3 πr3 r= radius of the sphere
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8.hemisphere 🌗 2πr2 3πr2 2/3πr3 r=radius of hemisphere
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9.spherical shell ⚽ .... 4 π(r2+r2) 4/3 π(R3-r3) R=external
radius
r=internal radius
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