Title: Basic Math Surface Area and Volume and Surface Area Formulas
1Basic Math Surface Area and Volume and Surface
Area Formulas
 Math for Water Technology
 MTH 082
 Lecture 3
 Chapters 9 10
2Objectives
 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
3RULES FOR AREA PROBLEMS
 IDENTIFY THE OBJECT
 LABEL THE OBJECT
 LOCATE THE FORMULA
 ISOLATE THE PARAMETERS NECESSARY
 CARRY OUT CONVERSIONS
 USE YOUR UNITS TO GUIDE YOU
 SOLVE THE PROBLEM
4What is surface area?
 Solid A 3D figure (combo of prism, clyinder,
cones, spheres, etc.)  Total surface area the sum of the areas of each
face of the 3D 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
5What is volume?
 The amount of space that a figure encloses
 It is threedimensional
 It is always answered in cubed units3
6Surface Area of a Sphere
 A sphere is a perfectly symmetrical,
threedimensional geometrical object all points
of which are equidistant from a fixed point.  Sphere Surface Area 4 p r² p d²
m
7The diameter of a sphere is 8 ft. What is the
ft2 surface area of the sphere?
8 ft
D8 ft A p d² A 3.14 (8 ft)2 A 3.14 (64
ft2) A 201 ft2
 31.4 ft2
 25.1 ft2
 201 ft2
 628 ft2
8Surface 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 its a vat.  Hemisphere or Vat Surface Area 2 p r²
d
r
9Volume of a Sphere and Hemisphere
 Sphere Volume 4 p r³ ( p d³)
 3
6  Hemisphere or VAT Volume (2 ) p r3

3
d
d
hemisphere
hemisphere
hemisphere
hemisphere
10The diameter of a sphere is 20 ft. What is the
ft3 volume of the sphere?
20 ft
D20 ft V 2 (0.785) (D2)(D) 3 V 2
0.785(20 ft)2(20 ft) 3 V
(12560 ft3) 3 V 4187 ft3
 526 ft3
 6583 ft3
 4187 ft3
 6280 ft3
11Volume of a cone
 Volume of cone 1/3 (p r² height)
 1/3 (¼ p 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/volumeformula.h
tml
12Volume of a cone
 Calculate the volume of a cone that is 3 m tall
and has a base diameter of 2m 
 V 1/3 (p r² height)
 V 1/3 (p 1m² 3m)
 V 1/3 (p 3 m3)
 V 1/3 (p 3 m3)
 V0.33(9.42m3)
 V3.14m3
3 m
2 m
13The 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?
D 50 ft, h 3 ft, I know r 25 ft! V 1
(0.785)(D2)h 3
V 1(0.785)(50 ft)2(3ft) 3 V
(0.785)(2500ft2)(3ft)
3 V(5888ft3) 3 V 1962ft3
h3 ft
r 25 ft
D 50 ft
 51032 ft3
 3533 ft3
 1649 ft3
 1962 ft3
14Lateral Surface Area of a Cone
 Area of cone 1/2 (p d slant height)

d diameter
slant height
15Cylinder (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.
16Volume of a Cylinder (TANK OR PIPE!!!)
 Volume p r² height ¼ p d²
height  Volume 0.785(diameter2)(depth)
17What 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?
D 75.2 ft, h 42.3 ft V 0.785(diameter2)(depth)
V(0.785)(75.2 ft)2(42.3ft) V
(0.785)(5655ft2)(42.3ft)
V(187,778ft3)
D75.2 ft
H42.3 ft
 2500 ft3
 188,000 ft3
 105,625 ft3
18A pipe is 16 inch in diameter and 550 ft long.
How many gallons does the pipe contain?
D 16 in or 1.33 ft, L 550 ft 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
D16 in
l550 ft
 4,295 gallons
 5,716 gallons
 51,670 gallons
 7,282 gallons
19Surface 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 (p 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 p r) h.  Add those two parts together and you have the
formula for the surface area of a cylinder
(www.webmath.com).
20Surface 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 pr2 2 prh
21Volume of water tank
 What is the volume of water contained in the tank
below if the side water depth is 12ft?
Volume p r² height Volume (3.14)
(5ft2)12 ft Volume (3.14)(25ft2)12 ft Volume
942 ft3
10 ft
16ftheight
12ftH20
22Surface Area of a Rectangular Prism
 In words, the surface area of a rectangular prism
is the area of the six rectangles that cover it.  a,b,c are the lengths
 Surface Area 2ab 2bc 2ac
cside
bside
aside
A2ab 2bc 2ac
23Volume of a rectangle (trench)
VL x W x H
24What 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?
L 245 ft, W 4.2 ft, D5.8 ft V L X W X H
V L X W X H
V 245 ft X 4.2 ft X 5.8 ft V 5968 or 6000 ft3
L245 ft
w4.2 ft
D5.8 ft
 51032 ft3
 1462209 ft3
 6000 ft3
25Volume 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 ftheight
VL x W x H V 10ft X 12 ftX10 ft V1,200 ft3
12ftwidth
10ftlength
26Volume of trough
bbase
Hheight
Llength
27Volume of water in trough
 Calculate the volume of water (in3) contained in
the trough if the water depth is 8 inches?
2 Ft24inches V(bh)(length)
2 V(4in)(8in)(24in) 2 V(32 in2)(24in)
2 V384 in3
4 inches
8 inches
2 ft
28Cylindrical Bottom tanks
A tank with a cylindrical bottom has dimensions
as shown below. What is the capacity of the
tank?
4 m
20 m
3 m
4 m
2 m
29Cylindrical Bottom tanks
4 m
3 m
2 m
30Cylindrical Bottom tanks
4 m
4 m
3 m
3 m
4 m
2 m
2 m
Representative Surface Area area of rectangle
area of half circle AL x w
(0.785)(d2)/2 A (4m)(3m)
0.785(4m2)/2 A 12m26.28m2
A18.28m2
Volume of tank area of surface x third
dimension V18.28m2 x 20m V356.6 m3
31Volume 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
lengthwidthheight
Volume Triangular prism
1/2lengthwidthheight
32Surface 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 onehalf the product of the perimeter
and the slant height.
L 0.5(P)(hs) Where P perimeter And Hs slant
height
hs
http//library.thinkquest.org/20991/geo/solids.htm
lpvolume
33Volume 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.
V1/3 L X W X H
http//library.thinkquest.org/20991/geo/solids.htm
lpvolume
34What 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?
35Review Surface Area Formulas!
 Sphere Surface Area 4 p r² p d²
 Hemisphere or Vat Surface Area 2 p r²
 Rectangular box surface area 2ab 2bc 2ac
 Surface Area Solid Cylinder 2 pr2 2 prh
 Surface Area Pyramid L 0.5(Perimeter)(slant
heighths)  Surface Area Prism
 (perimeter of shape b) L 2(Area of shape b)
36Review Volume Formulas!
 Sphere Volume 4/3 p r³ ( p d³)/6
 Hemisphere or VAT Volume (2/3) p r3
 Volume of Ellipsoid 4/3 p r1 r2 r3
 Volume of Cone 1/3 (p r² height)
1/3 (¼ p d² height)  Volume of Cylinder p r² height ¼
p 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
37Todays objective to become proficient with the
concept of volume as it pertains to water and
wastewater operation has been met
 Strongly Agree
 Agree
 Disagree
 Strongly Disagree