## 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

=2(1+b)*h+l*b

(v) Diagonal of the cuboid = √l^{2} + b^{2} + h^{2}

**SURFACE AREA OF A CUBE :** Consider a cube of edge a unit.

(i) The Total surface area of the cube = 6a^{2} sq. units

(ii) Lateral surface area of the cube = 4a^{2} 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 + 2πr^{2}

So total area of the cylinder=2πr(r + h)

**Remark :** Value of TE approximately equal to 22 / 7 or 3.14.

**APPLICATION:**

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

= 2πr_{1}h + 2πr_{2}h + 2π(r^{2}_{2} – r^{2}_{1})

= 2π(r_{1} + r_{2})h + 2π (r_{2} + r_{1}) (r_{2} – r_{1})

= 2π(r_{1} + r_{2}) [h + r_{2} – r_{1}]

**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 :** l^{2} = r^{2} + h^{2}

By applying Pythagorus

Theorem, here h is the height of the cone.

Thus l = √r^{2} + h^{2} and r = √l^{2} – h^{2}

h = √l^{2} + r^{2}

**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 * πr^{2}

= 4πr^{2}

Surface area ofa hemisphere = 2πr^{2}

Total surface area of a hemisphere = 2πr^{2} + πr^{2}

= 3πr^{2}

Total surface area of a hollow hemisphere with inner and outer radius r_{1} and r_{2} respectively

= 2πr^{2}_{1} + 2πr^{2}_{2} + π(r^{2}_{2} — r^{2}_{1})

= 2π(r^{2}_{1} + r^{2}_{2}) + π(r^{2}_{2} —r^{2}_{1})

**VOLUMES**

**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 = a^{3}

where a = edge of the cube

**VOLUME OF A CYLINDER**

Volume of a cylinder = πr^{2}h

volume of the hollow cylinder πr^{2}_{2}h — πr^{2}_{1}h

= π(r^{2}_{2} – r^{2}_{1})h

**VOLUME OF A RIGHT CIRCULAR CONE**

volume of a cone = 1 / 3 πr^{2}h, 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 πr^{3}, where r is the radius of the sphere.

Volume of a hemisphere = 2 / 3 πr^{3}

**APPLICATION :** Volume of the material of a hollow sphere with inner and outer radii r_{1} and r_{2} respectively

= 4 / 3 πr^{3}_{2} – 4 / 3 πr^{3}_{1} = 4 / 3π(r^{3}_{2} – r^{3}_{1})

Volume of the material of a hemisphere with inner and

outer radius r_{1} and r_{2} respectively = 2 / 3π(r^{3}_{2} – r^{3}_{1})