Basic Math Surface Area and
Volume and Surface Area
Formulas
Math for Water Technology
MTH 082
Lecture 3
Chapters 9 & 10
Objectives
•
•
Become proficient with the concept of
volume as it pertains to common
geometric shapes.
Solve waterworks math problems
equivalent to those on State of Oregon
Level I and Washington OIT Certification
Exams
RULES FOR AREA PROBLEMS
1. IDENTIFY THE OBJECT
2. LABEL THE OBJECT
3. LOCATE THE FORMULA
4. ISOLATE THE PARAMETERS NECESSARY
5. CARRY OUT CONVERSIONS
6. USE YOUR UNITS TO GUIDE YOU
7. SOLVE THE PROBLEM
What is surface area?
• Solid- A 3-D figure (combo of prism,
clyinder, cones, spheres, etc.)
• Total surface area- the sum of the areas of each
face of the 3-D solid
• Lateral surface area- The lateral area is the
surface area of a 3D figure, but excluding the
area of any bases (SIDES ONLY).
• It is always answered in square units2
• For example - to find the surface area of a cube
with sides of 5 inches, the equation is: Surface
Area = 6*(5 inches)2
= 6*(25 square inches)
= 150 sq. inches
What is volume?
• The amount of space that a figure encloses
• It is three-dimensional
• It is always answered in cubed units3
Surface Area of a Sphere
• A sphere is a perfectly symmetrical, three-dimensional
geometrical object; all points of which are equidistant
from a fixed point.
• Sphere Surface Area= 4 •π • r² = π • d²
m
u
f
c
r
D=diameter
r
e
r=radius
i
d
r
c
e
c
n
The diameter of a sphere is 8 ft.
What is the ft2 surface area of the
sphere?
8 ft
DRAW:
• Given:
D=8 ft
• Formula:
• Solve:
A= π • d²
9%
ft2
62
8
ft2
1
20
25
.1
ft2
0%
ft2
0%
.4
31.4 ft2
25.1 ft2
201 ft2
628 ft2
A= 3.14 (8 ft)2
A= 3.14 (64 ft2)
A= 201 ft2
31
1.
2.
3.
4.
91%
Surface Area of a Hemisphere
• A hemisphere is a sphere this is divided into two
equal hemispheres by any plane that passes through
its center; A half of a sphere bounded by a great
circle. In waterworks it’s a vat.
• Hemisphere or Vat Surface Area= 2 •π • r²
r
d
d
Vat
r
Volume of a Sphere and Hemisphere
• Sphere Volume = 4 • π • r³ = ( π• d³)
3
6
d
• Hemisphere or VAT Volume = (2 ) π r3
3
d
Vat
hemisphere
hemisphere
hemisphere
hemisphere
The diameter of a sphere is 20 ft.
What is the ft3 volume of the
sphere? DRAW:
100%
ft3
80
62
41
87
ft3
0%
ft3
83
65
526 ft3
6583 ft3
4187 ft3
6280 ft3
ft3
1.
2.
3.
4.
V= 2 (0.785) (D2)(D)
3
V= 2 * 0.785*(20 ft)2(20 ft)
3
V= (12560 ft3)
3
0%
0%
3
V= 4187 ft
6
• Solve:
D=20 ft
52
• Given:
• Formula:
20 ft
Volume of a cone
• Volume of cone = 1/3 (π • r² • height) =
1/3 (¼ • π • d² • height)
or
(0.785) (D²) (height)
3
A cone is a solid with a circular base. It has a curved surface which tapers
(i.e. decreases in size) to a vertex at the top. Cone height is the
perpendicular distance from the base to the vertex.
http://www.onlinemathlearning.com/volume-formula.html
Volume of a cone
• Calculate the volume of a cone that is 3 m tall and has
a base diameter of 2m
V= 1/3 (π • r² • height)
V= 1/3 (π • 1m² • 3m)
V= 1/3 (π • 3 m3)
3)
V=
1/3
(π
•
3
m
3m
V=0.33(9.42m3)
V=3.14m3
2m
The bottom portion of a tank is a cone. If the diameter
of the cone is 50 ft and the height is 3 ft, how many ft3
of water are needed to fill this portion of the tank?
• Given:
• Formula:
DRAW:
D= 50 ft, h= 3 ft, I know r= 25 ft!
h=3 ft
V= 1 (0.785)(D2)h
3
r= 25 ft
D= 50 ft
• Solve:
62
19
49
16
ft3
25%
ft3
25%
ft3
35
33
ft3
2
03
4. 1962 ft3
25% 25%
51
V= 1(0.785)(50 ft)2(3ft)
3
2)(3ft)
V=
(0.785)(2500ft
1. 51032 ft3
3
2. 3533 ft3 V=(5888ft3)
3
3
3. 1649 ft
V= 1962ft3
Lateral Surface Area of a Cone
• Area of cone = 1/2 (π • d • slant height) =
d= diameter
slant height
Cylinder
(TANK OR PIPE!!!)
A cylinder is a solid containing two parallel congruent circles. The cylinder
has one curved surface. The height of the cylinder is the perpendicular
distance between the two bases.
d=diameter
H=height
r=radius
Volume of a Cylinder
(TANK OR PIPE!!!)
• Volume = π • r² • height = ¼ • π • d² • height
• Volume= 0.785(diameter2)(depth)
d=diameter
H=height
r=radius
What is the capacity of a cylindrical tank in cubic feet if
it has a diameter of 75.2 ft and the height is 42.3 ft from
the base?
DRAW:
D= 75.2 ft, h= 42.3 ft
• Given:
• Formula:
V= 0.785(diameter2)(depth)
• Solve:
V=(0.785)(75.2 ft)2(42.3ft)
V= (0.785)(5655ft2)(42.3ft)
V=(187,778ft3)
10
5,
62
5
ft3
18
8,
00
0
00
ft3
0%
ft3
0%
25
1. 2500 ft3
2. 188,000 ft3
3. 105,625 ft3
100%
D=75.2 ft
H=42.3 ft
A pipe is 16 inch in diameter and 550 ft long. How many
gallons does the pipe contain?
V= 0.785(diameter2)(length)
V=(0.785)(1.33 ft)2(550 ft)
V= (0.785)(1.77ft2)(550 ft)
V= 764 ft3
V=(764ft3) (7.48 gal/1ft3)
V= 5716 gallons
ns
7,
28
2
ga
l
51
,6
70
6
71
5,
ga
llo
lo
n
s
ns
ga
llo
ns
ga
llo
5
29
4,295 gallons
5,716 gallons
51,670 gallons
7,282 gallons
25% 25% 25% 25%
4,
1.
2.
3.
4.
D= 16 in or 1.33 ft, L= 550 ft
D=16 in
• Given:
• Formula:
• Solve:
DRAW:
Surface Area of a Solid Cylinder
• In words, the easiest way is to think of a can. The surface
area is the areas of all the parts needed to cover the can.
That's the top, the bottom, and the paper label that wraps
around the middle.
• You can find the area of the top (or the bottom). That's the
formula for area of a circle (π r2). Since there is both a top
and a bottom, that gets multiplied by two.
• The side is like the label of the can. If you peel it off and lay
it flat it will be a rectangle. The area of a rectangle is the
product of the two sides. One side is the height of the can,
the other side is the perimeter of the circle, since the label
wraps once around the can. So the area of the rectangle is
(2 π r)* h.
• Add those two parts together and you have the formula for
the surface area of a cylinder (www.webmath.com).
Surface Area of a Solid Cylinder
• Surface Area = Areas of top and bottom +Area of the side
• Surface Area = 2(Area of top) + (perimeter of top)* height
• Surface Area = 2 πr2 + 2 πrh
d=diameter
H=height
r=radius
Volume of water tank
• What is the volume of water contained in the tank
below if the side water depth is 12ft?
10 ft
12ft=H20
16ft=height
Volume = π • r² • height
Volume (3.14) (5ft2)*12 ft
Volume= (3.14)(25ft2)*12 ft
Volume= 942 ft3
Surface Area of a Rectangular Prism
• In words, the surface area of a
rectangular prism is the area of the
six rectangles that cover it.
c=side
b=side
• a,b,c are the lengths
• Surface Area= 2ab + 2bc + 2ac
a=side
A=2ab + 2bc + 2ac
Volume of a rectangle (trench)
• Volume = L • W • H
l=length
h=height
w=width
V=L x W x H
What is the volume (ft3) of a trench in cubic feet if it has
a 245 ft length, 4.2 ft width, and 5.8 ft depth?
DRAW:
L= 245 ft, W= 4.2 ft, D=5.8 ft
• Given:
• Formula: V= L X W X H
• Solve:
L=245 ft
V= L X W X H
V= 245 ft X 4.2 ft X 5.8 ft
w=4.2 ft
D=5.8 ft
100%
V= 5968 or 6000 ft3
0%
ft3
00
60
20
62
14
03
2
9
ft3
ft3
0%
51
1. 51032 ft3
2. 1462209 ft3
3. 6000 ft3
Volume of water in tank
• Calculate the volume of water contained in the
rectangular tank. The depth to water with a side
water depth of 10 ft
– Volume = L • W • H
10 ft=height
V=L x W x H
V= 10ft X 12 ftX10 ft
V=1,200 ft3
Volume of trough
• Volume = (bh)(length)
2
b=base
H=height
L=length
Volume of water in trough
• Calculate the volume of water (in3) contained in the
trough if the water depth is 8 inches?
4 inches
8 inches
2 ft
2 Ft=24inches
V=(bh)(length)
2
V=(4in)(8in)(24in)
2
V=(32 in2)(24in)
2
V=384 in3
Cylindrical Bottom tanks
A tank with a cylindrical bottom has dimensions
as shown below. What is the capacity of the tank?
4m
20 m
3m
4m
2m
Cylindrical Bottom tanks
4m
20 m
3m
2m
4m
3m
2m
=
+
Cylindrical Bottom tanks
4m
3m
2m
4m
=
3m
+
4m
2m
Representative Surface Area = area of rectangle + area of half circle
A=L x w +(0.785)(d2)/2
A = (4m)(3m) +0.785(4m2)/2
A= 12m2+6.28m2
A=18.28m2
4m
Volume of tank = area of surface x third dimension
20 m
V=18.28m2 x 20m
3m
V=356.6 m3
2m
Volume of a Prism
• For the volume of any prism, then, you simply
determine the end area or the base area by the
appropriate method and multiply the end area by
the length or the base area by the height.
(b is the shape of the ends)
Volume rectangular prism=
length*width*height
Volume Triangular prism =
1/2*length*width*height
Surface Area of a Pyramid
• A regular pyramid is a pyramid that has a base
that is a regular polygon and with lateral faces that
are all congruent isosceles triangles
•
The area L of any regular pyramid with a base that has perimeter P and with slant height
hs is equal to one-half the product of the perimeter and the slant height.
hs
L =0.5(P)(hs)
Where P = perimeter
And Hs =slant height
http://library.thinkquest.org/20991/geo/solids.html#pvolume
Volume of a Pyramid
• A pyramid is a polyhedron with a single base and
lateral faces that are all triangular. All lateral
edges of a pyramid meet at a single point, or
vertex.
V=1/3 L X W X H
http://library.thinkquest.org/20991/geo/solids.html#pvolume
What did you learn?
• What is surface area?
• How are the units of surface area usually
expressed?
• What is volume?
• How many dimensions are in a volume
measurement?
• How are the units of volume usually expressed?
Review Surface Area Formulas!
•
•
•
•
•
•
Sphere Surface Area= 4 •π • r² = π • d²
Hemisphere or Vat Surface Area= 2 •π • r²
Rectangular box surface area= 2ab + 2bc + 2ac
Surface Area Solid Cylinder = 2 πr2 + 2 πrh
Surface Area Pyramid= L =0.5(Perimeter)(slant height=hs)
Surface Area Prism=
(perimeter of shape b) * L+ 2*(Area of shape b)
Review Volume Formulas!
Sphere Volume = 4/3 • π • r³ = ( π• d³)/6
Hemisphere or VAT Volume = (2/3) π r3
Volume of Ellipsoid= 4/3 • π • r1 • r2 • r3
Volume of Cone = 1/3 (π • r² • height) =
1/3 (¼ • π • d² • height)
Volume of Cylinder = π • r² • height = ¼ • π • d² • height
Volume of Rectangle or Rectangular prism = L • W • H
Volume of Triangular Prism= ½ L • W • H
Volume of trough = (bh)(length)
2
Today’s objective: to become proficient with
the concept of volume as it pertains to water
and wastewater operation has been met
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