J.N.V. NANDURBAR-2 MATHS PPT

Topic : Surface Area & Volume Name :Mr. Vinod Kumar Meena

1.SURFACE AREA

The surface area of a object is a measure of the total area that the surface of the object occupies .The mathematical definition of surface area in the presence of curved surfaces is consider by more involved than the definition of arc length of one- dimensional curves, or of the surface area for polyhedra, for which the surface area is the sum of the areas of its faces.

Smooth surfaces , such as a sphere are assigned surface area using their representation as parametric surface .

1.1 SURFACE AREA OF CUBOID

Surface area is the sum of the area of all faces or surfaces on a 3d shaped. A cuboid has 6 rectangular faces . We can also lable the length [L] , breadth,[B] and height [H] of the prism and use the formula,

SA= 2[LB +BH +HL]

1.2 SURFACE AREA OF CUBE

• A cube consist of ‘n’ number of square unit. Hence the space covered by these square unit on the surface of the cube is the surface area. Basically , the surface area is the sum of all the area of all the shapes that covered the surface of the shape or the object. In the case of a cube, there are six faces. So the surface area will be sum of all the area of six faces.
• Late us derive the formula for surface area for a given cube ,to solve problems based on it
• formulae =>as per the definition of cube, we know , the cube consist of six square faces let us consider , a cube whose length of edges is ‘a’.

• Now, we know , by the formula of the area of square ; area=side^2=a^2
• Therefore ,the total surface area of a cube

• = 6×a^2 = 6a^2

1.3 SURFACE AREA OF CYLINDER

• A cylinder, whose uniform cross section perpendicular to its height is a circle, is called a right circular cylinder.
• A right circular cylinder has two plane faces which are circular and curved surface.
• A right circular cylinder is a solid generated by the revolution of a rectangle about one of its sides.
• Here , the rectangle has been revolved about the side . The length of the side will be the height and the length of the side will be the radius of the cross section

Lateral surface area of a right circular cylinder

• Formula =>

Perimeter of the cross section * height

= 2πrh

Total surface area of right circular cylinder

• Formula => lateral surface area + 2*(area of the

cross section)

= 2πrh + 2πr^2

= 2πr(h + r)

1.4 SURFACE AREA OF CONE

• In order to calculate the curved surface area and total surface area of cone of a cone we divide into a circular base and the top slanted part . The area of the slanted part gives you the curved surface area. Total surface area is the sum of this circular base and curved surface area.
• Formula,

Curved surface area of cone = πrl Total surface area = πrl + πr^2

= πr(l + r)

1.5 Surface area of a Sphere

• The surface area of a sphere is defined as the amount of region covered by the surface of a sphere . A sphere is a 3-D solid object having around structure , just like a circle.
• Formula,

Surface area of Sphere = 4πr^2

= 2πr^2

(Curved surface area of Hemisphere ) (Total surface area of Hemisphere )

= 2πr^2 + πr^2

= 3πr^2

2. VOLUME

• Volume is the quantity of 3-D space enclosed by a close surface , for example the space that a substance or shape occupies or contains .
• The measure of space occupied by a solid body is called its volume .
• The unit of volume are cubic centimeter , liter , etc.

2.1 VOLUME OF CUBOID

• A cuboid is a 3-D structure having six rectangular faces

.This six faces of cuboid exists as a pair of 3 parallel faces .

• We mean to calculate the capacity of the box.
• The capacity of a cuboidal box is basically equal to the volume of cuboid involved.
• Formula,

volume of cuboid = length * breadth * height

= l * b * h

2.2 VOLUME OF CUBE

• The volume of figure is the number of cubes require to fill it completely , like blocks in box.
• Formula,

Volume of cube = side * side* side

= side^3

• Since each side of a square is the same , it can simply be the length of one side of cube.

2.3 VOLUME OF CYLINDER

• The volume of a cylinder is the density of the cylinder which signifies the amount of material it can carry or how much amount of any material can be immersed in it.
• Formula ,

Volume of Cylinder = πhr^2 r => radius of cross section h => height of cylinder

2.4 VOLUME OF A RIGHT CIRCULAR CONE

• The volume of a cone is one third of the product of the area of the base and the height of the cone.
• Formula ,

Volume of a right circular cone = (1/3)* πhr^2

2.5 VOLUME OF SPHERE

• Volume of sphere is defined as the capacity it has.
• Formula,

Volume of sphere = (4/3)* πr^2 Volume of Hemisphere = (2/3)* πr^2