# Understanding Squares

In geometry, a square is defined as a shape with four sides the same length (and with each side at a 90 degree angle to the others). But squares are very simple, which makes them a great way to start learning the basics of geometery.

For example, geometry makes it possible to calculate the area contained by a shape – and this is especially easy with a square. Since all four sides are the same length, the height of a square is the same as its length. Multiplying the height by itself will ultimately yield the same result as multiplying a square’s height by its length. It’s this fact which led ancient Greeks to refer to act of multiplying a number by itself as “squaring” that number.

For this reason three squares were often used to illustrate the Pythagorean theorem of the ancient Greek mathematician Pythagoras. A triangle was drawn where each side is also one of the four sides in a square. The area of each square was then calculated, revealing that the square touching the triangle’s hypotenuse has an area equal to the sum of the area of the other two squares.

Another principle of squares was also identified by Pythagoras. Squares can vary in size, but no matter how large or small their sides are, the diagonal of the square will always have exactly the same proportion to its sides. This number is the square root of two, and for this reason it’s known as Pythagoras’s constant.

But squares have other interesting properties. Adding the length of all four sides would give the perimeter – but since all four sides are the same length, this can also be obtained by simply multiplying the length of one side by four. And a diagonal line drawn from one corner across the square will have the same length as a diagonal line drawn from the other corner. One theorem even proved that the opposite was also true. A diamond can also have four sides which are all the same length – but if its two diagonal lines have the same length, then it must be a square, with its four sides all at 90-degree angles.

Other formulas identify the way a square relates to other geometric shapes. Placing a square at the center of a circle, so its sides touch the edges, will always leave the same proportion of circle space outside the square’s boundaries. This area will always equal one half the area of the square multiplied by pi – or about 1.57 times the square’s area. And drawing a diagonal across a square will always bisect it into two isosceles triangles, both with equal areas.

These formulas become especially handy when working with graphs, since the grid of points following an X-Y axis form a regular pattern of squares, and the area under a curve can often represented as a combination of squares and triangles. So ultimately understanding the formulas for measuring squares can be very useful in understanding the patterns on a graph.