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 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

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

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

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 Hemisphere or VAT Volume   =   (2 ) π r3 3 d d Vat hemisphere hemisphere hemisphere hemisphere

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

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

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

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. H=height r=radius d=diameter

Volume of a Cylinder (TANK OR PIPE!!!) Volume   =   π • r² • height   =   ¼ • π • d² • height Volume= 0.785(diameter2)(depth) H=height r=radius d=diameter

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

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

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   H=height r=radius d=diameter

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

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

Volume of a rectangle (trench) Volume   =   L • W • H w=width h=height l=length 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: 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 1462209 ft3 6000 ft3

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

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? 2 Ft=24inches V=(bh)(length) 2 V=(4in)(8in)(24in) V=(32 in2)(24in) V=384 in3 4 inches 8 inches 2 ft

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

Cylindrical Bottom tanks = + 2 m

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) +0.785(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

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. L =0.5(P)(hs) Where P = perimeter And Hs =slant height hs http://library.thinkquest.org/20991/geo/solids.html#pvolume

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 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 Strongly Agree Agree Disagree Strongly Disagree