Surface Area And Volume Of 3d Figures

PROJECT REPORT ON SURFACE AREA OF VOLUME OF 3D FIGURE Submitted by: S. Harpreet Singh Math Master

GOVT. GIRLS SEC. SCHOOL JANDIALA GURU, DISTT. AMRITSAR

SURFACE AREA An exterior surface of an object which appears to us is called its surface area.

What a Cylinder is? We come across round pillars, electric tube rods, water pipes, roller wires etc. in our

daily

life.

All

these

objects

are

cylindrical. If the circular ends are equal and there surfaces are parallel then such a cylinder is called a right circular cylinder.

Right Circular Cylinder If we round up a rectangular sheet of paper ABCD along its length or breadth i.e. A meets B and C meets D then the rectangle will become a figure as shown below: A

C

B

Rectangular Sheet

D

AB

CD Right Circular Cylinder

SURFACE AREA OF CYLINDER We have observed that a rectangular sheet can take the shape of cylinder. In this conversion we note that length of the rectangle becomes the perimeter of the circular base of the cylinder and its width becomes the height of the cylinder. h=b

b L C=l=2Πn

Thus we have: Height of cylinder h=b (breadth of rectangle) Perimeter of Circular base of Cylinder c=2Πr=l (length of rectangle) (IV)Curved surface area of cylinder = area of rectangle = l x b = 2Πrh (II) Total surface area of cylinder = area of curved surface + area of circular ends

base + circular top

= area of curved surface + area of circular area of = 2Πrh + Πr2 + Πr2 = 2Πrh + 2 Πr2

Volume The space occupied by slide is called its volume or the capacity of the container. Volume of Cylinder

= area of base of cylinder x height

Area of base of cylinder

= area of circular part of cylinder = Πr2

and direct height

= height of cylinder = h

Therefore of cylinder

= (Πr2 ) h = Πr2h (cubes of units of length)

Right Circular Cone A solid figure with its circular base and the corner (vertex) as a point is called a cone. The masson’s pendulum, cowdugs storage, loker’s cap, ice-cream cone are a few examples of cone with which we come across in our daily life. It the time loining the pointed edge to the centre of the base is normal to the circular base of cone, it is called right circular cone. Often we call the right circular cone just as cone.

Surface Area of the Cone Take a Circular piece of paper. Place the side OA on the side OB as shown: O 2Πr A

l

h

O

l r

l

A/B B

This becomes a Cone with the slant height l = radius of the circular segment of paper = OA or OB

Suppose r is the radius of the cone then area of the base = Πr2 Perimeter of base of cone = 2Πr (which is equal to the arc of the circular segment) arc of the circular segment = 2 Πr Because Area of the Circle

= Πr2 = 2Πr x r 2 = Perimeter x radius 2

Therefore of the circular segment

= Perimeter x radius 2 = 2Πr x l 2 = Πrl

•

Area of Curved Surface of Cone = 2Πr

•

Area of Total Surface of Cone

= Area of Curved surface + area of Base = Πrl + Πr2 = Πr (l + r)

h

l r

(iii) Slant height

= l =

√ h2

+ r2

Volume of Cone To find the volume of a cone: Take a cone and cylinder of Identical radius (r) and height (h). Fill the conical container with water and put this water in cylindrical container. Note that to fill the cylindrical container the process is repeated by how many times.

You observed that If we put three such completely tilled up conical containers in to the cylinder, it gets completely filled up. This is proved that 3 (Volume of Cone) = Volume of one full Cylindrical Container (Provided their r and h are equal) Therefore 3 x volume of cone = Πr2h Or volume of cone = 1 / 3 (Πr2h ) (Units of Volume)