Understanding the Mathematical Subject Squares

A square in plane geometry is a quadrilateral whose four sides are equal. The four equal sides determine four right angles. A right angle is an angle equal to 90 degrees. A square is a special kind of rectangle, which is any quadrilateral with four perpendicular sides. The four 90 degree angles of a square added together equal 360 degrees, the same number of degrees that are in a circle. A unit square is a square with all sides equal to one unit. Its perimeter is 1+1+1+1=4 and its area is 1^2=1.  Its graph is the segments formed by the coordinates (0,0), (0,1), (1,1), and (1,0).

The perimeter is the distance around an object. The perimeter of a square is found by adding the sides of the square together, or multiplying one side by 4. For example, the perimeter of a square with sides equal to 5 is 5+5+5+5=20, or 5×4=20. The area is the amount of space enclosed by an object. It is found by squaring one of the sides. In our example, the area is 5^2=25. One of the diagonals of a square makes two right triangles with the hypotenuse of both equal to the diagonal. Squares are two dimensional (length and width). A square has four parallel sides. Any positive integer squared is equal to the area of a square. Negative integers do not work because there is no such thing as a negative distance in plane geometry.

A chessboard or checkerboard is a square object because there are eight squares on each side. Parallelograms are four sided figures with parallel sides. A square is a parallelogram because the sides opposite each other are parallel. Squares are a part of Euclidean geometry, which includes straight lines and angles of plane geometry. One branch of Non-Euclidean geometry also includes curved lines, which can make almost any polygon a square. Triangles in Euclidean geometry have three sides. Trapezoids are four sided objects with two sides parallel and the other two not parallel. A rhombus is a parallelogram with all sides equal in length.

A magic square is a square that is composed of smaller squares, each of which has a number. The rows are the smaller squares that are in a horizontal line. The columns are the smaller vertical squares. The diagonals are the smaller squares that are located from the top left hand corner to the bottom right hand corner and the top right hand corner to the bottom left hand corner. The numbers in the smaller squares of the rows, columns, and diagonals all add up to the same number. This number is called the magic constant. For n=number of smaller squares, the equation for the magic constant is (n^3+n)/2. For example, the magic constant for 3 smaller squares is (3^3+3)/2=30/2=15 and for 4 smaller squares is (4^3+4)/2=(64+4)/2=34.