Whether you're buying carpet or paint, grass seed or fertilizer, roof shingles or paving stones, you need to know how to calculate the amount required. Buy too much, and you waste money. Buy too little, and you run out before the job is done. Being able to calculate area can save time and money but requires understanding what "square" means.

## Introducing Squares

To "square" means to calculate the value of a number multiplied by itself. A simple example is three squared, or three times three. Mathematically the problem looks like this: 3^{2} = 3 × 3 = 9. The exponent 2, written as superscript 2 (N^{2}), says to multiply a number (N) by itself, like so: N^{2} = N × N. Squared numbers always have the exponent or superscript of 2.

For large numbers, online calculator programs can be used. (See Resources)

## Calculating Area

To calculate area, multiply the length of the area by the width of the area. So, if carpet is needed for a 12-foot-long by 10-foot-wide room, simply multiply 12 × 10 to get 120 square feet, generally written as 120 ft^{2}. In the case of a 10-foot square room, since the length equals the width, the calculation becomes 10 × 10 = 10^{2} = 100 ft^{2}.

## Why Does Area Have Squared Units?

To help visualize area, use a piece of graph paper. Outline a rectangle four squares long by three squares wide. Count how many squares are contained within the outline. There are 4 × 3, or 12, squares contained within the outlined space. Area always has squared units, no matter what units (feet, meters, inches, etc.) have been measured.

## Converting From Square Inches to Square Feet

Remember that 12 linear inches equals 1 foot. On the graph paper, outline a space 12 squares long and 12 squares wide. Within that outlined square are 12^{2} or 12 × 12 = 144 smaller squares. So, 1 square foot contains 144 square inches.

To convert square inches to square feet requires dividing the area in square inches by 144 because 144 in^{2} equals 1 ft^{2}. So, if an area is 1440 in^{2}, but the paint container gives its coverage in terms of square feet, divide the 1440 in^{2} by 144 (because 144 in^{2} equals 1 ft^{2}) and find that the area 1440 in^{2} equals 10 ft^{2}. If a gallon of paint covers up to 400 square feet, then buying a pint of paint for this wall makes more economic sense.

If deciding whether to multiply or divide seems challenging, remember that there are 144 square inches in each square foot. The calculation from square inches to square feet should end up with a smaller number (division) while the calculation from square feet to square inches should end with a larger number (multiplication).

## Converting Square Feet to Square Yards

Converting square feet to square yards requires the same process. Returning to the graph paper, outline a three-by-three square (because 3 feet equals 1 yard). Counting the number of enclosed squares yields nine squares. Therefore, the conversion from square feet to square yards requires dividing by 9 while converting from square yards to square feet requires multiplying by 9.

## Converting a Square Meter to Square Feet

Since meters and feet come from different measurement systems, a conversion factor is necessary. Calculating based on 2.54 centimeters equaling 1 inch isn't complicated – just laborious – while looking up the conversion factor shows that 1 square meter (m^{2}) equals 10.764 square feet (ft^{2}). To change from square meters to square feet, multiply the number of square meters by 10.764 ft^{2} per m^{2}. To convert from square feet to square meters, divide by 10.764.

References

- Math Is Fun: Squares and Square Roots
- Home School Math: How to Calculate a Square Root Without a Calculator
- West Texas A&M University: Virtual Math Lab – Beginning Algebra Formulas
- University of Colorado – Boulder: Basic Geometry: Triangles, Circles, and More
- Math Open Reference: Area
- Southern Methodist University: Land Measurement Conversion Guide

About the Author

Karen earned her Bachelor of Science in geology. She worked as a geologist for ten years before returning to school to earn her multiple subject teaching credential. Karen taught middle school science for over two decades, earning her Master of Arts in Science Education (emphasis in 5-12 geosciences) along the way. Karen now designs and teaches science and STEAM classes.

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