# maths project surface area volume

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```					       Surface Area & Volume

OBJECTIVES
• Slices of 3-dimensional figures
• Nets of 3-D solids
• Lateral areas, surface areas,volumes of solids
• Properties of congruent & similar solids
3-dimensional figures
You will use different math websites for
visualizations of 3-D figures.
• Take time to use these sites to help explain the
material.
• Refer back to them for formulas, detailed
diagrams, and rotations
• Bookmark them or add to ‘favorites’ while you’re
in this course.
Websites for 3-dimensional geometry
http://www.mathsisfun.com/platonic_solids.html

http://www.mathsisfun.com/geometry/index.html

http://mathforum.org/alejandre/workshops/polyhedra.html

http://mathforum.org/library/drmath/sets/high_geom.html

http://www.houseof3d.com/pete/applets/graph/index.html
Nets & surface areas
• Nets are ‘flattened’ 3-D figures– it’s the pattern to
make the shape by folding.
• Refer to the text for vocabulary and diagrams as
well as the websites
• Add areas of all non-overlapping sections of the net
for each polygon for surface area of figure.
http://www.mathsisfun.com/platonic_solids.html
Surface area of prisms & cylinders
• Refer to the websites and your text for diagrams
and vocabulary: bases, lateral edges, altitude,
prism & cylinder
• Lateral area is the surface area minus the bases
• Surface area includes both bases
Right cylinder                  Regular prism
L.A. = Ph = 2 π r h                  L.A. = P h
S.A. = Ph+2π r2 = 2 π r h + 2π r2    S.A. = P h + 2B
Surface areas of pyramids & cones
http://www.mathsisfun.com/geometry/pyramids.html

• Refer to the websites and your text for diagrams
and vocabulary: base, vertex, lateral faces & edges,
altitude, and slant height.
• Refer to these resources for formulas
Volume of prisms & cylinders

• Right prisms:
Volume = B • h, where B = area of base & h = height

• Right cylinder:
Volume = B • h = π r 2 h
Volume of pyramids & cones
• Volume of Right pyramids:

1
V = B h, where B = area of base
3

• Volume of right cone:
V = 1 B h = 1 h r 2
3         3
Cavalarieri’s Principle: 2 solids of same height &
same cross-sectional area have the same volume.
Surface area & volume of spheres
• Refer to text for vocabulary

• Surface Area = 4 π r 2

4
• Volume =    r3
3
Congruent & similar solids
2 solids are congruent if:
corr 's  & corr edges 
areas of corr faces  & volumes 

2 solids are similar:
if ratio of 2 solids are a : b, then
surface area ratio – a 2 : b2
volume ratio – a 3 : b3
1.2 SURFACE AREA OF
A CUBOID & A CUBE ~
1 SURFACE AREA OF A CUBOID ~
The total surface area (TSA) of a cuboids is the
sum of the areas of its six faces. That is:
Example 1: Find the total surface area of a
cuboids with dimensions 8 cm by 6 cm by 5 cm.

Solution:
Surface area of a cube ~
To derive the formula of the surface area of a cube, start with a
cube as shown below and call the length of one side a:
In order to make a cube like the one
shown above, you basically use the
following cube template:

Looking at the cube template, it is easy to see that the cube
has six sides and each side is a square
The area of one square is a × a = a2
Example~2:
Find the surface area if the length of one side is
5 cm

SOL ~
Surface area = 6 × a2

Surface area = 6 × 42

Surface area = 6 × 4 × 4

Surface area = 96 cm2
SURFACE AREA OF A RIGHT CIRCULAR
CYLENDER ~
Curved Surface Area of a Cylinder
Recall:
Total Surface Area of a Cylinder ~

Consider a cylinder of radius r and height h.
The total surface area (TSA) includes the area of the circular top and
base, as well as the curved surface area (CSA)
The total surface area (TSA) of a cylinder with radius r and height h is
given by
Example 3 ~ Find the total surface area of a cylindrical
tin of radius 17 cm and height 3 cm.
Solution:
SURFACE AREA OF A RIGHT
CIRCULAR CONE ~
Total Surface Area of a Cone ~
Example 4~

Solution:
surface area = 4pr2

Example 5 :~ if 'r' = 5 for a given sphere, and p =
3.14, then the surface area of the sphere is:
surface area = 4pr2
= 4 × 3.14 × 52
= 314
Volume of a Cuboid
A cuboid with length l units, width w units and height h units has
a volume ofV cubic units given by
V = lwh
Example 5
Find the volume of a brick 30 cm by
25 cm by 10 cm.
Solution:
Volume of a Cylinder ~
A cylinder with radius r units and
height h units has
a volume of V cubic units given by
Example 6 ~ Find the volume of
a cylindrical canister with
radius 7 cm and height 12 cm.
Solution:
Volume of a Cone ~
The volume of a cone is given by
Example 42

Solution:
Volume of a Sphere ~
If four points on the surface of a sphere are joined to the
centre of the sphere, then a pyramid of perpendicular
height r is formed, as shown in the diagram. Consider the
solid sphere to be built with a large number of such solid
pyramids that have a very small base which represents a
small portion of the surface area of a sphere
Surface area and volume of different Geometrical Figures

Cube         Parallelopipe
Cylinder      Cone
d
Faces of cube

face

face
face

1
2   3

Dice (Pasa)

Total faces = 6 ( Here three faces are visible)
Faces of Parallelopiped

Face
Face

Face

Total faces = 6 ( Here only three faces are
visible.)

Book

Bric
k
Cores
Core
s

Total cores = 12 ( Here only 9 cores are visible)

Note Same is in the case in parallelopiped.
Surface area
Cube                                 Parallelopiped

c
a
b
a
a                                      Click to see the faces of
parallelopiped.
a
(Here all the faces are square)     (Here all the faces are rectangular)

Surface area = Area of all six        Surface area = Area of all six
faces                                 faces
= 6a2                     = 2(axb + bxc +cxa)
Volume of Parallelopiped   Click to animate

c

b
a                  b

Area of base (square) = a x b

Height of cube = c
Volume of cube = Area of base x
height x b) x c
= (a
Volume of Cube   Click to see

a                                 a

a

Area of base (square) =
a2
Height of cube = a
Volume of cube = Area of base x height
=       a2 x a       = a3(unit)   3
Outer Curved Surface area of cylinder               Click to animate

Activity -: Keep
bangles        of
r                                  h
one          over
another. It will
Circumference of          Formation of                     form a cylinder.
circle = 2 π r            Cylinder by bangles

It is the area covered by the outer surface of a cylinder.
Circumference of circle = 2 π r
Area covered by cylinder = Surface area of of cylinder = (2 π r)
Total Surface area of a solid cylinder

Curved
circular                       surface
surface
s

= Area of curved surface + area of two circular surfaces

=(2 π r) x( h) + 2 π r2

= 2 π r( h+ r)
Other method of Finding Surface area of cylinder with the help of
paper
r

h

h

2πr
Surface area of cylinder = Area of rectangle= 2 πrh
Volume of cylinder

r

h

Volume of cylinder = Area of base x vertical height
= π r2 xh
Cone

h
Base
r
Volume of a Cone              Click to See
the
experiment

h                                   h
Here the vertical height and
radius of cylinder & cone are
same.                                     r
r
3( volume of cone) = volume of cylinder
3( V )        = π r2 h
V = 1/3 π r2h
if both cylinder and cone have same height and radius then volume of a cylinder is three
times the volume of a cone ,

Volume = 3V                         Volume =V
Mr. Mohan has only a little jar of juice he wants to
distribute it to his three friends. This time he choose the
cone shaped glass so that quantity of juice seem to
appreciable.
Surface area of cone

l

2πr
l

l

Area of a circle having sector (circumference) 2π l = π l2πr
2

Area of circle having circumference 1 = π l 2/ 2 π l
Comparison of Area and volume of different geometrical figures

Surface   6a2         2π rh          πrl            4 π r2
area

Volume    a3          π r 2h         1/3π r2h       4/3 π r3
Area and volume of different geometrical figures

r
r
r/√2
r                        r                         l=2
r
Surface   6r2            2π r2         2π r2            2 π r2
area
=2 π r2

Volume    r3             3.14 r3       0.57π r3         0.47π r3
Total surface Area and volume of different geometrical figures and
nature
r
r
r
r              22r
1.44r                                               l=3
r
Total     4π r2      4π r2          4π r2                  4 π r2
Surface
area

Volume    2.99r3     3.14 r3        2.95 r3                4.18 r3

So for a given total surface area the volume of sphere is maximum.
Generally most of the fruits in the nature are spherical in nature
because it enables them to occupy less space but contains big amount
Think :- Which shape (cone or cylindrical) is better for collecting
resin from the tree

Click the
r                          r

3r

V= 1/3π r2(3r)                  V= π r2 (3r)
V= π r3
Long but Light in weight        V= 3 π r3
Long but Heavy in weight
Small niddle will
require to stick it             Long niddle
in the tree,so little           will require
harm in tree
to stick it in
the tree,so
Bottle

Cone
shape

Cylindrical
shape
If we make a cone having radius and height equal to the radius of
sphere. Then a water filled cone can fill the sphere in 4 times.

r
r
r

V=1/3 πr2h
V1
If h = r then
V=1/3 πr3
V1 = 4V = 4(1/3
πr3)
= 4/3 πr3
Volume of a Sphere            Click to See
the
experiment

h=
r          r
Here the vertical height and
radius of cone are same as
r   4( volume of cone) = volume of Sphere
4( 1/3πr2h ) = 4( 1/3πr3 ) = V
V = 4/3 π r3
Prashu
Kushagra roy
Kuldeep
Rishab Rawat
Vishal Lohan

U.C. Pandey R.C.Rauthan, G.C.Kandpal
Thanking you

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Description: maths project surface area volume