Calculation of area is the most basic and most important concept that must be understood in order to perform introductory-level water treatment math. Area calculations form the single building block upon which four of the six most important introductory-level math formulas are based. Area calulations are needed for such operations as calculating flow through filters, determining filter backwash rates, and calculating detention times in flocculation tanks and sedimentation basins.
On all introductory-level water treatment examinations in the United States, area is expressed in terms of inches and feet. The two key words, which indicate area and which indicate the scope of the area, are square and cubic. When area is expressed in inches, it will be designated as square inches or cubic inches. When area is expressed in feet, it will be designated as square feet or cubic feet.
A square measurement of area is always flat. To remember that the term square means flat, think of ordinary household carpeting, which is flat and which is usually measured in "square feet". A square foot is always 1 foot by 1 foot square. A square inch is always 1 inch by 1 inch square.
A cubic inch or cubic foot is always shaped like a perfect cube with the width, length, and height being exactly equal. A cubic inch is a cube that is 1 inch by 1 inch by 1 inch. A cubic foot is a cube that is 1 foot by 1 foot by 1 foot.
To obtain the square inches or square feet within a flat square or flat rectangular area, multiply the length of one side by the length of the other side. To obtain the cubic inches or cubic feet within a three-dimensional square or three-dimensional rectangular area, multiply the width of the area by the length of the area by the depth of the area. Below are several practice problems to illustrate this procedure.
1. A mixed media filter is 10 feet wide by 15 feet long. How many square feet are in the surface area of the filter? Answer: 10 feet x 15 feet = 150 square feet.
2. A sedimentation basin is 12 feet wide by 60 feet long by 30 feet deep. How many cubic feet of space is in the basin? Answer: 12 feet x 60 feet x 30 feet = 21,600 cubic feet
3. How many square inches are in a rectangle that is 3 feet by 5 inches? Answer: 36 inches x 5 inches = 180 square inches (Don't forget to convert the feet to inches.)
In addition to calculating square and cubic areas in straight-sided figures, introductory-level water treatment students must be able to calculate the area within a circle. Since a circle is always flat, versus a sphere which is three-dimensional, the area within a circle will always be expressed in the units that are used to express flat area - namely, SQUARE inches and SQUARE feet. Introductory-level water treatment students do not need to know the formula for calculating cubic inches and cublic feet within sphere-shaped objects.
The very clever formula for calculating area within a circle was devised over two thousand years ago by one of the greatest mathematicians in the ancient world. The simplified version of this formula, which all water treatment schools in the United States teach, is "D2 x .785". D2 is the diameter of the circle squared. To square a number, just multiply the number times itself.
The diameter of a circle might best be described as the full width of the circle. The radius of a circle is one-half of the full diameter, much like the spoke on a bicycle wheel is one-half the full width of the circular rim. Sometimes, water treatment math problems only give the length of a circle's radius. To obtain the circle's diameter, just multiply the radius by two.
There are three primary applications on introductory-level water treatment exams that require the use of the formula to calculate area within a circle. The first application involves the calculation of area within a circular basin, such as a solids-contact basin. The second application involves the calculation of area within a structure that has a round opening, such as a tube, hose, or pipe. The third application involves the calculation of area within a circular bulk storage tank or a circular day tank.
Below are two practice problems involving the area within a circle. Study these examples carefully.
1. A large circular basin is 32 feet across and 50 feet deep. How many cubic feet of water does the basin hold?
ANSWER: The first step is to find the area in square feet for the flat circle that forms the top surface of the basin. To do this, square the 32-foot diameter and multiply the answer by .785. (32 x 32 = 1024 x .785 = 803.84 square feet.) Next, multiply this answer by the depth of the basin to obtain the three-dimensional cubic area within the basin. 803.84 x 50 = 40,192 cubic feet.
2. How many cubic inches of water does a 100-foot length of 8-inch pipe hold?
ANSWER: First, find the flat area that forms the surface opening of the 8-inch pipe. (8 x 8 = 64 x .785 = 50.24 square inches.) Next, because inches can only be multipled by inches, convert the 100-foot length to inches. (100 x 12 inches = 1200 inches.) Now, multiply the flat surface area by the three-dimensional length. (50.24 square inches x 1200 inches of depth = 60,288 cubic inches.) The 100-foot segment of 8-inch pipe in the question will hold 60,288 cubic inches of water.