1. Basic Math Surface Area and Volume and Surface Area Formulas
Math for Water Technology
MTH 082
Lecture 3
Chapters 9 & 10

2. 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

3. RULES 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

4. 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       

5. What is volume? The amount of space that a figure encloses
It is three-dimensional
It is always answered in cubed units3

6. 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
D=diameter
r=radius c i r u f e n r d

7. The diameter of a sphere is 8 ft
The diameter of a sphere is 8 ft. What is the ft2 surface area of the sphere?
DRAW:
Given:
Formula:
Solve:
8 ft
D=8 ft
A= π • 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

8. 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

9. Volume of a Sphere and Hemisphere
Sphere Volume   =   4 • π • r³  = ( π• d³)
Hemisphere or VAT Volume   =   (2 ) π r3 3 d d
Vat
hemisphere
hemisphere
hemisphere
hemisphere

10. The diameter of a sphere is 20 ft. What is the ft3 volume of the sphere?
DRAW:
Given:
Formula:
Solve:
20 ft
D=20 ft
V= 2 (0.785) (D2)(D) 3
V= 2 * 0.785*(20 ft)2(20 ft)
V= (12560 ft3)
V= 4187 ft3
526 ft3
6583 ft3
4187 ft3
6280 ft3

11. 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.

12. 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) 
V= 1/3 (π • 3 m3)
V=0.33(9.42m3)
V=3.14m3
3 m
2 m

13. The bottom portion of a tank is a cone
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?
DRAW:
Given:
Formula:
Solve:
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)
V= (0.785)(2500ft2)(3ft)
V=(5888ft3)
V= 1962ft3
h=3 ft
r= 25 ft
D= 50 ft
51032 ft3
3533 ft3
1649 ft3
1962 ft3

14. Lateral Surface Area of a Cone
Area of cone   =   1/2 (π • d • slant height)  =
d= diameter
slant height

15. 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.
H=height
r=radius
d=diameter

16. Volume of a Cylinder (TANK OR PIPE!!!)
Volume   =   π • r² • height   =   ¼ • π • d² • height
Volume= (diameter2)(depth)
H=height
r=radius
d=diameter

17. 2500 ft3 188,000 ft3 105,625 ft3 DRAW: Given: Formula: Solve:
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:
Given:
Formula:
Solve:
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)
D=75.2 ft
H=42.3 ft
2500 ft3
188,000 ft3
105,625 ft3

18. 4,295 gallons 5,716 gallons 51,670 gallons 7,282 gallons DRAW: Given:
A pipe is 16 inch in diameter and 550 ft long. How many gallons does the pipe contain?
DRAW:
Given:
Formula:
Solve:
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
D=16 in
l=550 ft
4,295 gallons
5,716 gallons
51,670 gallons
7,282 gallons

19. 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 (

20. 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  
H=height
r=radius
d=diameter

21. Volume of water tank
What is the volume of water contained in the tank below if the side water depth is 12ft?
Volume   =   π • r² • height
Volume (3.14) (5ft2)*12 ft
Volume= (3.14)(25ft2)*12 ft
Volume= 942 ft3
10 ft
16ft=height
12ft=H20

22. 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.
a,b,c are the lengths
Surface Area= 2ab + 2bc + 2ac
c=side
b=side
a=side
A=2ab + 2bc + 2ac

23. Volume of a rectangle (trench)
Volume   =   L • W • H
w=width
h=height
l=length
V=L x W x H

24. 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:
Given:
Formula:
Solve:
L= 245 ft, W= 4.2 ft, D=5.8 ft
V= L X W X H
V= 245 ft X 4.2 ft X 5.8 ft
V= 5968 or 6000 ft3
L=245 ft
w=4.2 ft
D=5.8 ft
51032 ft3
ft3
6000 ft3

25. Volume of water in tank V=L x W x H V= 10ft X 12 ftX10 ft V=1,200 ft3
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
12ft=width
10ft=length

26. Volume of trough
Volume = (bh)(length) 2
b=base
H=height
L=length

27. Volume of water in trough
Calculate the volume of water (in3) contained in the trough if the water depth is 8 inches?
2 Ft=24inches
V=(bh)(length) 2
V=(4in)(8in)(24in)
V=(32 in2)(24in)
V=384 in3
4 inches
8 inches
2 ft

28. Cylindrical 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

29. Cylindrical Bottom tanks= +
2 m

30. Cylindrical Bottom tanks=
3 m +
4 m
2 m
2 m
Representative Surface Area = area of rectangle + area of half circle
A=L x w +(0.785)(d2)/2
A = (4m)(3m) (4m2)/2
A= 12m2+6.28m2
A=18.28m2
2 m
3 m
20 m
4 m
Volume of tank = area of surface x third dimension
V=18.28m2 x 20m
V=356.6 m3

31. 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

32. 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.
L =0.5(P)(hs)
Where P = perimeter
And Hs =slant height hs

33. Volume of a Pyramid V=1/3 L X W X H
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

34. 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?

35. 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)

36. 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)  = /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

37. Today’s 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