Basic Math for the Small Public Water Systems Operator

1
Basic Math for the Small Public Water Systems
Operator

• Small Public Water Systems Technology Assistance
  Center
• Penn State Harrisburg

2
Introduction

• Area
• In this module we will learn how to calculate the
  area of some basic shapes that include the
• Rectangle,
• Triangle, and
• Circle

3
Overview

• Calculating the area of a basic shape is a
  necessary step in determining the volume or
  capacity of a container.
• Being able to calculate the surface area of a
  tank has practical applications as well. For
  example, knowing the surface area of a tank will
  enable you to estimate the quantity of paint
  required to paint that tank.

4
Basic Shapes

• Rectangle
• Triangle
• Circle Cylinder

5
Area Calculations

• Area calculations are two dimensional. They
  involve two dimensions such as length and width.
• For example when we multiply the linear unit feet
  times the linear unit feet we get the area unit
  measurement of square feet.

6
Area Calculations

• So the unit multiplication ft x ft gives the
  answer ft or sq ft.
• An example in the Metric system of measurement
  would be to multiply the linear unit meter times
  the linear unit meter for a result of m or sq m.

2
2
7
Calculating the Area of a Rectangle

• The formula to calculate the area of a rectangle
  is
• Area (Length)(Width)
• or
• A (L)(W)

Width
Length
8
Example - Calculating the Area of a Rectangle

• Calculate the area of a rectangle whose length is
  25 feet and whose width is 15 feet.
• Area Length (Feet) x Width (Feet)
• Area 25 ft x 15 ft
• Area 375 sq ft

15 ft
25 ft
9
Practice Exercise

• 1. Calculate the area of a rectangle whose
  length is 50 feet and whose width is 30 feet.

Answer 1,500 sq ft
10
Solution

• Area Length (Feet) x Width (Feet)
• Area 50 ft x 30 ft
• Area 1,500 ft2

11
Practice Exercise

• 2. Calculate the area of a rectangle whose
  length is 42 feet and whose width is 23 feet.

Answer 966 sq ft
12
Solution

• Area Length (Feet) x Width (Feet)
• Area 42 ft x 23 ft
• Area 966 ft2

13
Calculating the Area of a Triangle

• The formula to calculate the area of a triangle
  is
• Area (Base)(Height)
• 2
• or
• A (B)(H)
• 2

Height
Base
14
Example Calculating the Area of a Triangle

• Calculate the area of a triangle whose
• base is 16 feet and whose height is 32 feet.
• Area (Square Feet) Base (Feet) x Height (Feet)
• 
  2
• Area 16 ft x 32 ft
• 2
• Area 256 sq ft

15
Practice Exercise

• 1. Calculate the area of a triangle whose base
  is 60 feet and whose height is 120 feet.

Answer 3,600 sq ft
16
Solution

• Area (Base)(Height)
• 2
• Area 60 ft x 120 ft
• 2
• Area 3,600 ft2

17
Practice Exercise

• 2. Calculate the area of a triangle whose base
  is 54 feet and whose height is 152 feet.

Answer 4,104 sq ft
18
Solution

• Area (Base)(Height)
• 2
• Area 54 ft x 152 ft
• 2
• Area 4,104 ft2

19
Calculating the Circumference of a Circle

• The circumference of a circle is the distance
  around the circle.
• The formula to calculate the circumference of
• 
  C ? x D
• Where ? (pronounced pi)
• is the Greek symbol for the value 3.14
  and D is the
  diameter.

Diameter
20
Example Calculating the Circumference of a
Circle

• Calculate the circumference of a circle whose
  diameter is 3 feet.
• Circumference 3.14 x
  3 ft
• Circumference 9.42 ft

21
Practice Exercise

• 1. Calculate the circumference of a circle whose
  diameter is 5 feet.

Answer 15.7 ft
22
Solution

• Circumference ? x D
• C ? x 5 ft
• C 3.14 x 5 ft
• C 15.7 ft

23
Practice Exercise

• 2. Calculate the circumference of a circle whose
  diameter is 25 feet.

Answer 78.5 ft
24
Solution

• C ? x D
• C ? x 25 ft
• C 3.14 x 25 ft
• C 78.5 ft

25
Calculating the Area of a Circle

• The formula to calculate the area of a circle is
• Area ? x r
• Where ? (pronounced pi)
• is the Greek symbol for the value 3.14
  and r is the
• radius squared.

2
Diameter
Radius
26
Relationship of the Radius to the Diameter of a
Circle

• The diameter of a circle is two times the radius.
• Diameter 2 x
  Radius
• 
  or
• D
  2 x r

27
Example Calculating the Area of a Circle

• Calculate the area of a circle whose radius is 4
  feet.
• Area ? x r2
• Area 3.14 x (4 ft)2
• Area 3.14 x 16 sq ft
• Area 50.27 sq ft

28
Practice Exercise

• 1. Calculate the area of a circle whose radius
  is 5 feet.

Answer 78.54 sq ft
29
Solution

• Area ? x r2
• Area 3.14 x (5 ft)2
• Area 78.5 ft2

30
Practice Exercise

• 2. Calculate the area of a circle whose diameter
  is 50 feet. Hint The diameter divided in half
  is equal to the radius.

Answer 1,963.50 sq ft
31
Solution

• Area ? x r2
• Area 3.14 x (25 ft)2
• Area 1,963.5 ft2

32
Calculating the Surface Area of a Cylinder

• To calculate the surface area break the cylinder
  down into its component parts. That is two
  circles and its wall.

Circumference ? x Diameter
Height
33
Surface Area of a Cylinder

• We already know how to calculate the area of a
  circle by applying the formula
• Area ? x r2
• Remember the cylinder is comprised of
  two circles, therefore it is
  necessary to multiply the above formula by 2.

34
Surface Area of a Cylinder

• To calculate the area of the cylinder wall, first
  calculate its length by using the following
  formula
• Area ? x D
• Where D is the diameter of the circle.
• Next multiply this result by the height of the
  tank.

35
Surface Area of a Cylinder

• Finally, add the area of the two circles and the
  area of the tank wall to obtain the total surface
  area of the tank.

36
Example Calculating the Surface Area of a
Cylinder

• Calculate the surface area of a tank with a
  radius of 35 feet and a height of 45 feet.
• First Calculate the area of the tank top and
  bottom as follows
• Area 2 x ? x r2
• Area 2 x 3.14 x (35 ft)2
• Area 7,697 sq ft

37
Example Calculating the Surface Area of a
Cylinder

• Next Calculate the length of the tank wall as
  follows
• Length ? x D
• Length 3.1416 x 70 ft
• Length 220 ft
• Remember, the diameter is found by multiplying
  the radius by 2.

38
Example Calculating the Surface Area of a
Cylinder

• Next Multiply the length of the tank wall by
  the height of the tank to obtain the area of the
  tank wall
• Area Length x Height
• Area 220 ft x 45 ft
• Area 9,896 sq ft

39
Example Calculating the Surface Area of a
Cylinder

• Finally, add the area of the tank top and bottom
  together with the area of the tank wall to obtain
  the total surface area of the tank.
• 7,697 sq ft 9,896 sq ft 17,593 sq ft

40
Practice Exercise

• 1. Calculate the surface area of a tank with a
  diameter of 20 feet and a height of 40 feet.

Answer 3,142 sq ft
41
Solution

• Area of tank top and bottom
• 2 x ? x r2
• 2 x 3.14 x (10 ft)2 628.32 ft2
• Length of tank wall
• ? x Diameter
• ? x 20 ft 62.83 ft

42
Solution Continued

• Area of tank wall
• Length x Height
• 62.83 ft x 40 ft 2,513.27 ft2
• Total area of tank
• 628.32 ft2 2,513.27 ft2 3,142 ft2

43
Practice Exercise

• 2. Calculate the surface area of a tank with a
  diameter of 15 feet and a height of 20 feet.

Answer 1,296 sq ft
44
Solution

• Area of tank top and bottom
• 2 x ? x r2
• 2 x 3.14 x (7.5 ft)2 353.43 ft2
• Length of tank wall
• ? x Diameter
• ? x 15 ft 47.12 ft

45
Solution Continued

• Area of tank wall
• Length x Height
• 47.12 ft x 20 ft 942.48 ft2
• Total area of tank
• 353.43 ft2 942.48 ft2 1,296 ft2

46
Summary

• At the completion of this training module you
  should be able to calculate the area of the three
  basic shapes introduced the rectangle, triangle
  and the circle.
• The next module demonstrates how to expand upon
  area calculations to determine volumes of various
  types of tanks, which are components of our water
  treatment systems.