Maths Class 9 Notes for Volume and Surface Area
SOLIDS : The bodies occupying space (i.e. have 3-dimension) are called solids such as a cuboid, a cube, a cylinder, a cone, a sphere etc.
VOLUME (CAPACITY) OFA SOLID: The measure of space occupied by a solid-body is called its volume. The units of volume are cubic centimeters (written as cm3) or cubic meters (written as m3).
CUBOID: A solid bounded by six rectangular faces is called a cuboid.
In the given figure, ABCDEFGH is a cuboid whose
(i) 6 faces are :
ABCD, EFGH, ABFE, CDHQ ADHE, and BCGF Out of these, the four faces namely ABFE, DCGH, ADHE and BCGF are called lateral faces of the cuboid.
(ii) 12 edges are :
AB, BC, CD, DA, EF, FG GH, HE, CG BF, AE and DH
(iii) 8 vertices are :
A, B, C, D, E, F, and H.
Remark : A rectangular room is in the form of a cuboid and its 4 walls are its lateral surfaces.
Cube : A cuboid whose length, breadth and height are all equal, is called a cube.
A cube has 6 faces, each face is square, 12 edges, all edges are of equal lengths and 8 vertices.
SURFACE AREA OF A CUBOID:
Let us consider a cuboid of length = 1 units
Breadth = b units and height = h units
Then we have :
(i) Total surface area of the cuboid
=2(l * b + b * h + h * l) sq. units
(ii) Lateral surface area of the cuboid
= [2 (1 + b)* h] sq. units
(iii) Area of four walls of a room = [2 (1 + b)* h] sq. units.
= (Perimeter of the base * height) sq. units
(iv) Surface area of four walls and ceiling of a room
= lateral surface area of the room + surface area of ceiling
(v) Diagonal of the cuboid = √l2 + b2 + h2
SURFACE AREA OF A CUBE : Consider a cube of edge a unit.
(i) The Total surface area of the cube = 6a2 sq. units
(ii) Lateral surface area of the cube = 4a2 sq. units.
(iii) The diagonal of the cube = √3 a units.
SURFACE AREA OF THE RIGHT CIRCULAR CYLINDER
Cylinder: Solids like circular pillars, circular pipes, circular pencils, road rollers and gas cylinders etc. are said to be in cylindrical shapes.
Curved surface area of the cylinder
= Area of the rectangular sheet
= length * breadth
= Perimeter of the base of the cylinder * height
= 2πr * h
Therefore, curved surface area of a cylinder = 2πrh
Total surface area of the cylinder =2πrh + 2πr2
So total area of the cylinder=2πr(r + h)
Remark : Value of TE approximately equal to 22 / 7 or 3.14.
If a cylinder is a hollow cylinder whose inner radius is r1 and outer radius r2 and height h then
Total surface area of the cylinder
= 2πr1h + 2πr2h + 2π(r22 – r21)
= 2π(r1 + r2)h + 2π (r2 + r1) (r2 – r1)
= 2π(r1 + r2) [h + r2 – r1]
SURFACE AREA OF A RIGHT CIRCULAR CONE
RIGHT CIRCULAR CONE
A figure generated by rotating a right triangle about a perpendicular side is called the right circular cone.
SURFACE AREA OF A RIGHT CIRCULAR CONE:
curved surface area of a cone = 1 / 2 * l * 2πr = πrl
where r is base radius and l its slant height
Total surface area of the right circular cone
= curved surface area + Area of the base
= πrl + πr2 = πr(l + r)
Note : l2 = r2 + h2
By applying Pythagorus
Theorem, here h is the height of the cone.
Thus l = √r2 + h2 and r = √l2 – h2
h = √l2 + r2
SURFACE AREA OF A SPHERE
Sphere: A sphere is a three dimensional figure (solid figure) which is made up of all points in the space which lie at a constant distance called the radius, from a fixed point called the centre of the sphere.
Note : A sphere is like the surface of a ball. The word solid sphere is used for the solid whose surface is a sphere.
Surface area of a sphere: The surface area of a sphere of radius r = 4 x area of a circle of radius r = 4 * πr2
Surface area ofa hemisphere = 2πr2
Total surface area of a hemisphere = 2πr2 + πr2
Total surface area of a hollow hemisphere with inner and outer radius r1 and r2 respectively
= 2πr21 + 2πr22 + π(r22 — r21)
= 2π(r21 + r22) + π(r22 —r21)
VOLUME OF A CUBOID :
Volume : Solid objects occupy space.
The measure of this occupied space is called volume of the object.
Capacity of a container : The capacity of an object is the volume of the substance its interior can accommodate.
The unit of measurement of either of the two is cubic unit.
Volume of a cuboid : Volume of a cuboid =Area of the base * height V=l * b * h
So, volume of a cuboid = base area * height = length * breadth * height
Volume of a cube : Volume of a cube = edge * edge * edge = a3
where a = edge of the cube
VOLUME OF A CYLINDER
Volume of a cylinder = πr2h
volume of the hollow cylinder πr22h — πr21h
= π(r22 – r21)h
VOLUME OF A RIGHT CIRCULAR CONE
volume of a cone = 1 / 3 πr2h, where r is the base radius
and h is the height of the cone.
VOLUME OF A SPHERE
volume of a sphere the sphere = 4 / 3 πr3, where r is the radius of the sphere.
Volume of a hemisphere = 2 / 3 πr3
APPLICATION : Volume of the material of a hollow sphere with inner and outer radii r1 and r2 respectively
= 4 / 3 πr32 – 4 / 3 πr31 = 4 / 3π(r32 – r31)
Volume of the material of a hemisphere with inner and
outer radius r1 and r2 respectively = 2 / 3π(r32 – r31)