Square

A square is a special kind of shape in geometry. It is a regular quadrilateral, meaning it has four straight sides that are all the same length. Each angle in a square is a right angle, which measures 90 degrees or π/2 radians. The sides of a square are always perpendicular to each other.

Squares are very useful in many areas of life. They appear in tiled floors and walls, graph paper, the tiny squares that make up digital images called pixels, and many game boards. You can also find squares in building floor plans, pieces of paper used for folding like origami paper, servings of food, and in art and designs.

In math, the area of a square is found by multiplying the length of one side by itself. Squares can be used to solve interesting problems, like trying to draw a square inside a circle using only a compass and straightedge.

Definitions and characterizations

A square is a special shape with four equal sides and four right angles. This means it is both a rectangle and a rhombus. There are many ways to describe a square, and if a shape meets any one of these rules, it will meet them all. For example, a square can be a rectangle with all sides equal, or a rhombus with a right angle between two sides. Squares are the only regular shapes where all the angles inside and outside the shape are the same, and they are all right angles.

Properties

A square is a special shape with four equal sides and four equal angles, each being 90 degrees. It is a type of rectangle and rhombus.

Squares are all the same shape, but they can be different sizes. If you know the length of one side, you can find out everything about the square. When squares have the same size, they are called congruent.

For a square with side length ℓ, the perimeter is 4ℓ and the diagonal length is √2 ℓ. The area of the square is ℓ².

The square has many symmetries. It can be turned by 90°, 180°, or 270° and still look the same. It can also be flipped across its diagonals or sides.

A square can fit inside a circle, and a circle can pass through all four corners of the square. The circle inside the square touches all four sides, and the circle outside the square touches all four corners.

Applications

Squares are very useful shapes in many areas of life. They are commonly used as the shape of tiles, like in floors and walls. The word for a small tile, "tessera," even comes from an ancient word meaning the number four, because of the four corners of a square tile.

In technology, squares are important too. Graph paper with a grid of squares helps with drawing and measuring. The tiny dots, called pixels, in digital photos and videos are arranged in a square grid. This square pattern helps in storing and showing images on screens. Many modern buildings, from ancient pyramids to modern skyscrapers, have square designs. Artists and designers often use squares in their work for balance and order. Games like chessboard and sports like baseball diamonds also use square shapes. Even some foods, like certain types of waffles, are square. Squares appear in nature and science too, in patterns and molecular structures.

Constructions

Coordinates and equations

A unit square is a square with sides of length one. You can see it on a grid where the points have coordinates between 0 and 1 on both the x and y axes. The four corners of this square are at points where the x and y coordinates are either 0 or 1.

In the world of complex numbers, multiplying a number by the imaginary unit i turns it 90 degrees. If you take a complex number and multiply it by i four times, you get four new numbers that form the corners of a square centered at the origin. You can move this square to any other center point by adding another complex number.

Compass and straightedge

You can draw a square with a given side length using just a compass and straightedge, as described in Euclid's Elements. This shows that squares are constructible shapes.

Related topics

The cube and regular octahedron are next in sequences of regular polytopes that start with squares.

The Sierpiński carpet is a square fractal with square holes.

The Schläfli symbol of a square is { 4 } {\displaystyle \{4\}} . A truncated square is an octagon. The square is part of a group of regular polytopes that includes the cube in three dimensions and the hypercubes in higher dimensions. It is also part of another group that includes the regular octahedron in three dimensions and the cross-polytopes in higher dimensions.

The Sierpiński carpet is a square fractal with square holes. Space-filling curves such as the Hilbert curve, Peano curve, and Sierpiński curve can cover a square.

Many theorems include squares. Cross's theorem or Vecten's theorem says that for a triangle made from the sides of three squares, connecting the squares' points to make three more triangles, all triangles have the same area.

Puzzles with squares include square arrays filled with numbers in Sudoku with different symbols in Latin square, and with color or blank in nonogram. There are also optical illusions like the missing square puzzle and the chessboard paradox. The Langton's ant, in cellular automaton, moves on a grid of squares.

A square trisection is a problem where a square is cut into pieces and rearranged into three identical squares. Tarski's circle-squaring problem asks if a circle can be cut into pieces and rearranged into a square.

Inscribed squares

Main articles: Inscribed square problem and Inscribed square in a triangle

A square is inscribed in a curve when all four points of the square are on the curve. The inscribed square problem asks if every simple closed curve has an inscribed square. This is true for every smooth curve and for any closed convex curve. The only other regular shape that can always be inscribed in every closed convex curve is the equilateral triangle.

For an inscribed square in a triangle, at least one side of the square lies on a side of the triangle. Every acute triangle has three inscribed squares. A right triangle has two inscribed squares. An obtuse triangle has only one inscribed square. A square inscribed in a triangle can cover at most half the triangle's area.

Area and quadrature

See also: Area, Quadrature (geometry), and Squaring the circle

Since ancient times, many units of area have been based on squares, often using a standard length as the side, like a square meter or square inch.

In ancient Greek deductive geometry, the area of a shape was found by making a square of the same area using only a few steps with a compass and straightedge, called quadrature or squaring. Euclid's Elements shows how to do this for rectangles, parallelograms, triangles, and other simple polygons.

Because squares were used to measure area, the Greeks and later mathematicians tried unsuccessfully to square the circle, making a square with the same area as a circle using only a few steps with a compass and straightedge.

Tiling and packing

Square tiling

Pythagorean tiling

The square tiling, seen in floors and game boards, is one of three regular tilings of the plane. The other two use the equilateral triangle and the regular hexagon. The points of a square tiling form a square lattice. Squares of different sizes can also tile the plane, as in the Pythagorean tiling, named for its link to proofs of the Pythagorean theorem.

Square packing problems ask for the smallest square or circle that can hold a certain number of unit squares.

Squaring the square means dividing a square into smaller squares, all with whole number side lengths. A division with all different smaller squares is called a perfect squared square.

Counting

A common mathematical puzzle is counting all the squares in a square grid of n × n {\displaystyle n\times n} squares. For example, a grid of nine squares has 14 squares in total. The answer is n ( n + 1 ) ( 2 n + 1 ) / 6 {\displaystyle n(n+1)(2n+1)/6} , a square pyramidal number.

Another counting problem asks for the number of different rectangle shapes that can fit when dividing a square into similar rectangles.

A magic square is a square grid of numbers where the sums of the numbers in each row, each column, and both main diagonals are the same.

Other geometries

Concentric squares in the sphere (orthographic projection)

Concentric squares in the hyperbolic plane (conformal disk model)

In Euclidean geometry, space is flat, and every four-sided shape has angles adding to 360°, so a square has four equal sides and four right angles (each 90°). In spherical geometry and hyperbolic geometry, space is curved, and four-sided shapes with four right angles do not exist. Both geometries have regular four-sided shapes with four equal sides and four equal angles, often called squares.

In spherical geometry, space curves in a way that makes the angles of a four-sided shape add up to more than 360°. Small spherical squares are close to Euclidean squares, but larger ones have bigger angles.

In hyperbolic geometry, space curves so that the angles of a four-sided shape add up to less than 360°. Small hyperbolic squares are close to Euclidean squares, but larger ones have smaller angles.

The Euclidean plane can be described using the real coordinate plane with the Euclidean distance rule. Other geometries use different distance rules, and in some of these, shapes that would be Euclidean squares become "circles". Squares turned at 45° are the circles in taxicab geometry. In this geometry, points at a set distance form a diagonal square. In another geometry, axis-parallel squares act like circles.