Basic Math for the Small Public Water Systems Operator

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Basic Math for the Small

Public Water Systems

Operator

Small Public Water Systems

Technology Assistance Center

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Introduction

• Area

• In this module we will learn how to

calculate the area of some basic shapes that include the:

Rectangle,

Triangle, and

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• 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.

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Rectangle

Triangle

Circle Cylinder

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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.

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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

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• The formula to calculate the area of a rectangle is:

Area = (Length)(Width) or

A = (L)(W)

Calculating the Area of a

Rectangle

W

idth

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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

25 ft

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Practice Exercise

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

50 ft

30 ft

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Solution:

• Area = Length (Feet) x Width (Feet)

• Area = 50 ft x 30 ft

• Area = 1,500 ft

2

50 ft

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Practice Exercise

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

Answer: 966 sq ft

23 ft 42 ft

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Solution:

• Area = Length (Feet) x Width (Feet)

• Area = 42 ft x 23 ft

• Area = 966 ft

2

23 ft 42 ft

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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

He

ig

ht

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• 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 32 ft

16 ft

Example – Calculating the Area

of a Triangle

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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

60 ft

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Solution:

• Area = (Base)(Height) 2

• Area = 60 ft x 120 ft 2

• Area = 3,600 ft2

60 ft

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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

54 ft

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Solution:

• Area = (Base)(Height) 2

• Area = 54 ft x 152 ft 2

• Area = 4,104 ft2 54 ft

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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.

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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

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Practice Exercise

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

5 ft

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Solution:

• Circumference =  x D

• C =  x 5 ft

• C = 3.14 x 5 ft

• C = 15.7 ft

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Practice Exercise

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

Answer: 78.5 ft

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Solution:

• C =  x D

• C =  x 25 ft

• C = 3.14 x 25 ft

• C = 78.5 ft

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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.

Diameter

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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

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• 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 = 50.24 sq ft

r

4 ft

Example – Calculating the Area

of a Circle

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

Practice Exercise

r

5 ft

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Solution:

• Area =  x r2

• Area = 3.14 x (5 ft)2

• Area = 78.5 ft2

r

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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: 1,963.5 sq ft

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Solution:

• Area =  x r2

• Area = 3.14 x (25 ft)2

• Area =1,963.5 ft2

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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

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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

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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.

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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.

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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

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Example – Calculating the

Surface Area of a Cylinder

• Next: Calculate the length of the tank wall as follows:

Length =  x D

Length = 3.14 x 70 ft Length = 220 ft

• Remember, the diameter is found by multiplying the radius by 2.

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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

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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.

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Practice Exercise

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

20 ft

40 ft

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Solution:

•Area of tank top and bottom: •2 x  x r2

•2 x 3.14 x (10 ft)2 = 628 ft2

•Length of tank wall:

• x Diameter

• x 20 ft = 62.8 ft

20 ft

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Solution Continued

• Area of tank wall: • Length x Height

• 62.8 ft x 40 ft = 2,512 ft2

• Total area of tank:

• 628 ft2 + 2,512 ft2 = 3,140 ft2

20 ft

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Practice Exercise

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

Answer: 1,295.25 sq ft

15 ft

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Solution:

• Area of tank top and bottom: • 2 x  x r2

• 2 x 3.14 x (7.5 ft)2 = 353.25 ft2

• Length of tank wall:

•  x Diameter

•  x 15 ft = 47.1 ft

15 ft

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Solution Continued

• Area of tank wall: • Length x Height

• 47.1 ft x 20 ft = 942 ft2

• Total area of tank:

• 353.25 ft2 + 942 ft2 = 1,295.25 ft2

15 ft

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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.