Basic Math for the Small Public Water Systems Operator

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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 2 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 2 for a result of m or sq m.

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. 50 ft 30 ft Answer: 1,500 sq ft

10 Solution: 50 ft 30 ft Area = Length (Feet) x Width (Feet) Area = 50 ft x 30 ft Area = 1,500 ft 2

11 Practice Exercise 2. Calculate the area of a rectangle whose length is 42 feet and whose width is 23 feet. 42 ft 23 ft Answer: 966 sq ft

12 Solution: 42 ft 23 ft Area = Length (Feet) x Width (Feet) Area = 42 ft x 23 ft Area = 966 ft 2

13 Height Calculating the Area of a Triangle The formula to calculate the area of a triangle is: Base Area = (Base)(Height) 2 or A = (B)(H) 2

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 32 ft Area = 256 sq ft 16 ft

15 Practice Exercise 1. Calculate the area of a triangle whose base is 60 feet and whose height is 120 feet. 60 ft 120 ft Answer: 3,600 sq ft

16 Solution: Area = (Base)(Height) 2 Area = 60 ft x 120 ft 2 Area = 3,600 ft 2 60 ft 120 ft

17 Practice Exercise 2. Calculate the area of a triangle whose base is 54 feet and whose height is 152 feet. 54 ft 152 ft Answer: 4,104 sq ft

18 Solution: Area = (Base)(Height) 2 Area = 54 ft x 152 ft 2 Area = 4,104 ft 2 54 ft 152 ft

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 Diameter C = x D Where (pronounced pi) is the Greek symbol for the value 3.14 and D is the diameter.

20 Example Calculating the Circumference of a Circle Calculate the circumference of a circle whose diameter is 3 feet. D = 3 ft 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. 5 ft Answer: 15.7 ft

22 Solution: Circumference = x D C = x 5 ft C = 3.14 x 5 ft C = 15.7 ft 5 ft

23 Practice Exercise 2. Calculate the circumference of a circle whose diameter is 25 feet. 25 ft Answer: 78.5 ft

24 Solution: C = x D C = x 25 ft C = 3.14 x 25 ft C = 78.5 ft 25 ft

25 Calculating the Area of a Circle The formula to calculate the area of a circle is: Diameter Area = x r Where (pronounced pi) is the Greek symbol for the value 3.14 and r is the radius squared. 2

26 Relationship of the Radius to the Diameter of a Circle The diameter of a circle is two times the radius. Diameter 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. r 4 ft Area = x r 2 Area = 3.14 x (4 ft) 2 Area = 3.14 x 16 sq ft Area = sq ft

28 Practice Exercise 1. Calculate the area of a circle whose radius is 5 feet. r 5 ft Answer: 78.5 sq ft

29 Solution: Area = x r 2 Area = 3.14 x (5 ft) 2 Area = 78.5 ft 2 r 5 ft

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: 1,963.5 sq ft 50 ft

31 Solution: Area = x r 2 Area = 3.14 x (25 ft) 2 50 ft Area =1,963.5 ft 2

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 r 2 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 r 2 Area = 2 x 3.14 x (35 ft) 2 Area = 7,693 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 = 3.14 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 = 9,900 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. 7,693 sq ft + 9,900 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. 20 ft 40 ft Answer: 3,140 sq ft

41 Solution: Area of tank top and bottom: 2 x x r 2 2 x 3.14 x (10 ft) 2 = 628 ft 2 Length of tank wall: x Diameter x 20 ft = 62.8 ft 20 ft 40 ft

42 Solution Continued 20 ft Area of tank wall: Length x Height 62.8 ft x 40 ft = 2,512 ft 2 Total area of tank: 628 ft 2 + 2,512 ft 2 = 3,140 ft 2 40 ft

43 Practice Exercise 2. Calculate the surface area of a tank with a diameter of 15 feet and a height of 20 feet. 15 ft 20 ft Answer: 1, sq ft

44 Solution: Area of tank top and bottom: 2 x x r 2 2 x 3.14 x (7.5 ft) 2 = ft 2 Length of tank wall: x Diameter x 15 ft = 47.1 ft 15 ft 20 ft

45 Solution Continued Area of tank wall: Length x Height 47.1 ft x 20 ft = 942 ft 2 Total area of tank: ft ft 2 = 1, ft 2 15 ft 20 ft

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.

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