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

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

Basic Math Surface Area and Volume and Surface Area

Formulas

Basic Math Surface Area and Volume and Surface Area

Formulas

Math for Water TechnologyMTH 082Lecture 3

Chapters 9 & 10

Math for Water TechnologyMTH 082Lecture 3

Chapters 9 & 10

Page 2: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

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

• 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

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

RULES FOR AREA PROBLEMSRULES FOR AREA PROBLEMS1. 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

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

Page 4: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

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    

  

• 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    

  

Page 5: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

What is volume?

• The amount of space that a figure encloses

• It is three-dimensional

• It is always answered in cubed units3

• The amount of space that a figure encloses

• It is three-dimensional

• It is always answered in cubed units3

Page 6: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Surface Area of a SphereSurface 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

• 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

mm

D=diameterD=diameter

r=radiusr=radius

ccii

rr

ccuu ff

rr

ee

nncc

eerd

Page 7: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

DRAW:• Given:

• Formula:• Solve:

DRAW:• Given:

• Formula:• Solve:

The diameter of a sphere is 8 ft. What is the ft2 surface area of the

sphere?

31.

4 ft2

25.

1 ft2

201

ft2

628

ft2

0%9%

91%

0%

D=8 ft

A= π • d²

A= 3.14 (8 ft)2

A= 3.14 (64 ft2)A= 201 ft2

D=8 ft

A= π • d²

A= 3.14 (8 ft)2

A= 3.14 (64 ft2)A= 201 ft2

1. 31.4 ft2

2. 25.1 ft2

3. 201 ft2

4. 628 ft2

1. 31.4 ft2

2. 25.1 ft2

3. 201 ft2

4. 628 ft2

8 ft

Page 8: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Surface Area of a HemisphereSurface 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 

• 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

rd

VatVat

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

Volume of a Sphere and HemisphereVolume of a Sphere and Hemisphere• Sphere Volume   =   4 • π • r³  = ( π• d³)

3 6

• Hemisphere or VAT Volume   =   (2 ) π r3

3

• Sphere Volume   =   4 • π • r³  = ( π• d³)

3 6

• Hemisphere or VAT Volume   =   (2 ) π r3

3

VatVat

d

d

hemispherehemisphere hemispherehemisphere

hemispherehemisphere

hemispherehemisphere

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

The diameter of a sphere is 20 ft. What is the ft3 volume of the

sphere?

526

ft3

658

3 ft3

418

7 ft3

628

0 ft3

0% 0%

100%

0%

D=20 ft

V= 2 (0.785) (D2)(D) 3V= 2 * 0.785*(20 ft)2(20 ft) 3V= (12560 ft3) 3V= 4187 ft3

D=20 ft

V= 2 (0.785) (D2)(D) 3V= 2 * 0.785*(20 ft)2(20 ft) 3V= (12560 ft3) 3V= 4187 ft3

1. 526 ft3

2. 6583 ft3

3. 4187 ft3

4. 6280 ft3

1. 526 ft3

2. 6583 ft3

3. 4187 ft3

4. 6280 ft3

20 ft

DRAW:• Given:• Formula:

• Solve:

DRAW:

• Given:• Formula:

• Solve:

Page 11: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of a coneVolume of a cone• Volume of cone   =   1/3 (π • r² • height)  =

1/3 (¼ • π • d² • height)

or

(0.785) (D²) (height)

3

• 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.

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

Page 12: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of a coneVolume of a cone• Calculate the volume of a cone that is 3 m tall and has

a base diameter of 2m• Calculate the volume of a cone that is 3 m tall and has

a base diameter of 2m

2 m2 m

3 m3 m

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

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

Page 13: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

DRAW:•Given:•Formula:

•Solve:

DRAW:•Given:•Formula:

•Solve:

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?

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?

510

32 ft

3

353

3 ft3

164

9 ft3

196

2 ft3

25% 25%25%25%

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) 3V=(5888ft3) 3V= 1962ft3

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) 3V=(5888ft3) 3V= 1962ft3

D= 50 ft

r= 25 fth=3 ft

1. 51032 ft3

2. 3533 ft3

3. 1649 ft3

4. 1962 ft3

1. 51032 ft3

2. 3533 ft3

3. 1649 ft3

4. 1962 ft3

Page 14: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

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

slant heightslant height

d= diameterd= diameter

Page 15: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Cylinder (TANK OR PIPE!!!)

Cylinder (TANK OR PIPE!!!)

H=

heig

ht

H=

heig

ht

r=radius

d=diameter

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.

Page 16: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of a Cylinder (TANK OR PIPE!!!)

Volume of a Cylinder (TANK OR PIPE!!!)

• Volume   =   π • r² • height   =   ¼ • π • d² • height• Volume= 0.785(diameter2)(depth)

• Volume   =   π • r² • height   =   ¼ • π • d² • height• Volume= 0.785(diameter2)(depth)

H=

heig

ht

H=

heig

ht

r=radius

d=diameter

Page 17: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

DRAW:•Given:•Formula:•Solve:

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?

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?

250

0 ft3

188

,000

ft3

105

,625

ft3

0% 0%

100%

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

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 ftD=75.2 ft

H=42.3 ftH=42.3 ft

1. 2500 ft3

2. 188,000 ft3

3. 105,625 ft3

1. 2500 ft3

2. 188,000 ft3

3. 105,625 ft3

Page 18: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

DRAW:•Given:•Formula:•Solve:

DRAW:•Given:•Formula:•Solve:

A pipe is 16 inch in diameter and 550 ft long. How many gallons does the pipe contain?

A pipe is 16 inch in diameter and 550 ft long. How many gallons does the pipe contain?

4,2

95 g

allo

ns

5,7

16 g

allo

ns

51,

670

gallo

ns

7,2

82 g

allo

ns

25% 25%25%25%

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 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 inD

=16 in

l=550 ftl=550 ft

1. 4,295 gallons

2. 5,716 gallons

3. 51,670 gallons

4. 7,282 gallons

1. 4,295 gallons

2. 5,716 gallons

3. 51,670 gallons

4. 7,282 gallons

Page 19: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Surface Area of a Solid CylinderSurface 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).

• 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).

Page 20: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Surface Area of a Solid CylinderSurface 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  

• 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=

heig

ht

H=

heig

ht

r=radius

d=diameter

Page 21: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of water tankVolume of water tank• What is the volume of water contained in the tank

below if the side water depth is 12ft?• What is the volume of water contained in the tank

below if the side water depth is 12ft?

16ft=h

eigh

t16ft=

heig

ht

10 ft10 ftVolume   =   π • r² • heightVolume (3.14) (5ft2)*12 ftVolume= (3.14)(25ft2)*12 ftVolume= 942 ft3

Volume   =   π • r² • heightVolume (3.14) (5ft2)*12 ftVolume= (3.14)(25ft2)*12 ftVolume= 942 ft3

12ft=H

2 012ft=

H2 0

Page 22: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

• 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

• 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

Surface Area of a Rectangular PrismSurface Area of a Rectangular Prism

a=sidea=side

b=sideb=side

A=2ab + 2bc + 2ac A=2ab + 2bc + 2ac

c=sidec=side

Page 23: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of a rectangle (trench)Volume of a rectangle (trench)

• Volume   =   L • W • H• Volume   =   L • W • H

w=widthw=width

h=heighth=height

l=lengthl=length

V=L x W x H V=L x W x H

Page 24: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

DRAW:•Given:•Formula:•Solve:

DRAW:•Given:•Formula:•Solve:

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?

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?

510

32 ft

3

146

2209

ft3

600

0 ft3

0%

100%

0%

L= 245 ft, W= 4.2 ft, D=5.8 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

L= 245 ft, W= 4.2 ft, D=5.8 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

L=245 ft

w=4.2 ft

D=5.8 ft

1. 51032 ft3

2. 1462209 ft3

3. 6000 ft3

1. 51032 ft3

2. 1462209 ft3

3. 6000 ft3

Page 25: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of water in tankVolume 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

• 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

10ft=length

10ft=length

10 ft=height10 ft=height

12ft=width12ft=width V=L x W x H

V= 10ft X 12 ftX10 ftV=1,200 ft3

V=L x W x HV= 10ft X 12 ftX10 ftV=1,200 ft3

Page 26: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of troughVolume of trough• Volume = (bh)(length)

2

• Volume = (bh)(length) 2

L=lengthL=length

H=heightH=height

b=baseb=base

Page 27: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of water in troughVolume of water in trough• Calculate the volume of water (in3) contained in the

trough if the water depth is 8 inches?• Calculate the volume of water (in3) contained in the

trough if the water depth is 8 inches?

2 ft2 ft

8 inches8 inches

4 inches4 inches

2 Ft=24inchesV=(bh)(length) 2V=(4in)(8in)(24in)

2V=(32 in2)(24in)

2V=384 in3

2 Ft=24inchesV=(bh)(length) 2V=(4in)(8in)(24in)

2V=(32 in2)(24in)

2V=384 in3

Page 28: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Cylindrical Bottom tanksCylindrical Bottom tanks

2 m

3 m

20 m

4 m

A tank with a cylindrical bottom has dimensions as shown below. What is the capacity of the tank? A tank with a cylindrical bottom has dimensions as shown below. What is the capacity of the tank?

4 m

Page 29: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Cylindrical Bottom tanksCylindrical Bottom tanks

2 m

3 m

20 m

4 m

4 m

2 m

3 m= +

Page 30: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

4 m4 m

2 m2 m

3 m3 m== ++

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

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

4 m4 m

3 m3 m

2 m2 m

4 m4 m

Volume of tank = area of surface x third dimension V=18.28m2 x 20m V=356.6 m3

Volume of tank = area of surface x third dimension V=18.28m2 x 20m V=356.6 m3

2 m

3 m20 m

4 m

Cylindrical Bottom tanksCylindrical Bottom tanks

Page 31: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of a PrismVolume 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.

• 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) (b is the shape of the ends)

Volume Triangular prism = 1/2*length*width*height Volume Triangular prism = 1/2*length*width*height

Volume rectangular prism= length*width*heightVolume rectangular prism= length*width*height

Page 32: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Surface Area of a PyramidSurface 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.

• 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.

http://library.thinkquest.org/20991/geo/solids.html#pvolume

L =0.5(P)(hs)Where P = perimeterAnd Hs =slant height

L =0.5(P)(hs)Where P = perimeterAnd Hs =slant height

hs

Page 33: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Volume of a PyramidVolume 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.

• 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 HV=1/3 L X W X H

http://library.thinkquest.org/20991/geo/solids.html#pvolume

Page 34: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

What did you learn?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?

• 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?

Page 35: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

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)

• 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)

Page 36: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

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

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

Page 37: Basic Math Surface Area and Volume and Surface Area Formulas Math for Water Technology MTH 082 Lecture 3 Chapters 9 & 10 Math for Water Technology MTH

Today’s objective: to become proficient with the concept of volume as it pertains to water

and wastewater operation has been met

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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1. Strongly Agree

2. Agree

3. Disagree

4. Strongly Disagree

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2. Agree

3. Disagree

4. Strongly Disagree