Square in Math: Definition & Overview

Instructor: Betty Bundly

Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.

In this lesson, we will learn about a very common concept in mathematics. As we will see, the concept of square can be found in algebra and geometry. Furthermore, it is also an often-encountered measurement.

What Is Square?

When someone says 'square,' this can have different meanings in math, but they all point to the same basic mathematical concept. When you square something, it is multiplied by itself. For example, 3 squared means 3x3. This common operation is notated with an exponent of 2. Using this notation, 3x3 is indicated with 3^2 and is read as '3 squared' or '3 to the second power' or '3 to the power of 2'. Squaring is not just limited to numbers, however. You can square variables or even complex mathematical expressions.

Whenever a quantity is squared, the result will always be positive. If you square a number, the answer is always positive because (-)*(-) = +. For example, (3)^2 = (3)*(3) = 9 and (-3)^2 = (-3)*(-3) = 9.

Square in Geometry

Square also has a specific meaning in geometry. A square is a flat or 2-dimensional shape with the following properties:

It has four sides.

All four sides have the same length.

The angle between any two sides next to each other (also called adjacent sides) is 90°. This means that all four angles in a square have a measure of 90°.

Square diagram

You may wonder how this is related to the concept of multiplying something by itself. The relationship comes from the area of a square, which is a measurement of the space within the four sides of the square. To find the area of a square, you must multiply the length of one side of the square by the length of another side of the square, and since all four sides are equal, that's the same as multiplying the length of one side of the square by itself. If s represents the length of one side of the square, then the area of a square is given by the formula Area=s^2.

As an example of using this formula to calculate area, suppose you want to buy a curtain for a square-shaped window and need to know what size curtain to purchase. If one side of the window is measured to be 1.5 feet, then the area of the window is 1.5 feet x 1.5 feet = 2.25 square feet and you would need a square shaped curtain measuring at least 2.25 square feet. This area could also be expressed as 1.5 feet by 1.5 feet, as the word 'by' is often used instead of the multiplication symbol to describe an area. Also, notice that to calculate the area, you squared the unit of measurement, feet, along with squaring the number 1.5. Again, numbers are not the only quantities that can be squared.

Square Units in Measurement

All area is measured in square units. Some common units of measurement are square inches, square feet or square miles. In metric measurements, commonly seen units are square millimeters, square centimeters, square meters, or square kilometers. As we know, area measurements are made every day in all kinds of shapes that aren't squares. Then, you may ask, how is it that these areas are all given in square units? You could think of it this way - if you cut the shape into a lot of small squares and each equals one square unit in measurement, you could then add all the small one-unit sized squares. The sum of these small unit squares would equal the total number of square units and that is essentially how shapes that aren't squares are measured in square units.

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