Square

A square has four straight sides.

All four sides are the same length. It also has four equal corners.

These corners are called right angles.

Squares are very special. They are a type of rectangle. They are also a type of rhombus.

Many things use this shape. You can see them on floor tiles. They are on game boards too. You can find them on graph paper.

Squares can fit together perfectly. They can cover a whole floor. This is called a tiling.

Can you spot a square near you?

A square is a special shape with four straight sides.

All four sides are the same length. The four corners are also equal. We call these right angles.

Because it has right angles, a square is a type of rectangle. Because its sides are equal, it is also a type of rhombus.

We can measure a square in many ways. The area is the space inside the shape. To find it, multiply the side length by itself. This math task is called squaring.

You can also find the perimeter. This is the total distance around the outside.

Squares are very useful in our world. They fit together perfectly to cover floors or walls. This is called tiling. You see squares in many places. They are in pixels on your screen. They are on game boards and graph paper. Even some buildings use square floor plans.

Ancient pyramids often had square bases. Squares are everywhere!

A square is a very special kind of shape.

It belongs to a group of shapes called quadrilaterals, which means it has four sides. To be a square, all four sides must be the exact same length. The four corners must also be equal. We call these corners right angles, which are 90 degrees.

Because it has right angles, a square is a type of rectangle. Because its sides are equal, it is also a type of rhombus. This makes the square one of the most balanced shapes in geometry.

We can use math to measure many things about a square. The area is the amount of space inside the shape. You find the area by multiplying the side length by itself.

This math task is so common that we call it "squaring" a number. The perimeter is the distance all the way around the outside edges. If you have a square with sides of length $s$, the perimeter is $4s$. You can also find the length of the diagonal line that cuts through the middle. The diagonal of a square is the side length multiplied by the square root of 2.

People have studied squares for a very long time.

Ancient mathematicians in Babylon were already calculating square diagonals between 1800 and 1600 BCE. They knew that the square root of 2 is an irrational number. This means it cannot be written as a simple fraction. It is about 1.414. There are also many hard puzzles involving squares. Some mathematicians study how to divide a square into smaller, unequal squares. Others try to figure out how to pack squares tightly into other shapes.

Squares are very useful for organizing our world.

They can tile a flat surface perfectly without leaving any gaps. You see this in tiled floors, walls, and graph paper. Digital images use tiny squares called pixels to make pictures. Even many buildings use square floor plans to stay practical. Ancient Egyptian pyramids and East Asian pagodas often have square bases. Some sports areas, like boxing rings, use a square shape too.

Because they are so balanced, squares have many symmetries.

You can rotate a square or flip it in different ways, and it will still look exactly the same. There are eight different ways to move a square so it fits perfectly back into its original spot. This makes the square the most symmetrical of all the four-sided shapes. Squares also relate to circles in interesting ways. You can fit a circle perfectly inside a square so it touches every side. You can also draw a circle that passes through all four corners of the square.

In geometry, a square is a specific type of regular quadrilateral. A quadrilateral is any polygon with four sides. To be a regular quadrilateral, a shape must have equal side lengths and equal interior angles. A square meets both criteria perfectly. It has four straight sides of identical length and four right angles, which measure exactly 90 degrees or $\pi/2$ radians. Because its angles are 90 degrees, the adjacent sides are perpendicular to one another.

A square is a unique shape because it belongs to several different families of polygons. It is a special case of a rectangle, which is defined by having four right angles. It is also a special case of a rhombus, which is defined by having four equal sides. Because it fits both definitions, a square is both a rectangle and a rhombus. This relationship means a square inherits all the properties of these shapes. For example, it is also a parallelogram, a kite, a trapezoid, and a tetragon.

Mathematicians can define a square in many equivalent ways using different geometric properties. You might describe it as a rectangle with four equal sides. Alternatively, you could call it a rhombus that contains at least one right angle. Another way to identify a square is by looking at its diagonals. In a square, the diagonals are equal in length and are perpendicular bisectors of each other. This means they cross at a 90-degree angle and cut each other exactly in half. The diagonals also bisect the internal angles, creating two 45-degree angles at each corner.

Measurement and algebra are closely tied to the properties of the square. The area of a square is calculated by multiplying the side length by itself. This specific operation is so fundamental that mathematicians use the term "squaring" to describe raising any number to the second power.

If a square has a side length of $s$, its perimeter is $4s$. The length of its diagonal is $s\sqrt{2}$. The value of $\sqrt{2}$ is an irrational number, meaning it cannot be expressed as a simple fraction. It is approximately 1.414. This value was known to Babylonian mathematicians as far back as 1800 to 1600 BCE.

Squares possess a high degree of symmetry, more than any other quadrilateral. There are eight rigid transformations that can move a square so that it looks exactly as it did before. These include various rotations around the center and reflections across different axes.

These eight symmetries form a mathematical structure known as the dihedral group of order eight. This group includes combinations of rotations and reflections. When you perform these moves, the square always maps onto itself. This high level of symmetry is why squares are often used as the basic unit in repeating patterns, known as wallpaper groups.

In practical applications, squares are incredibly useful for organization and design. Because equal squares can tile a plane edge-to-edge, they are used for tiled floors, walls, and graph paper. In the digital world, the pixels in bitmap images are often arranged in a square grid.

Even complex data structures, like quadtrees used in image compression, rely on dividing squares into smaller squares. Architecture also frequently uses square footprints. Ancient structures like the Egyptian pyramids and East Asian pagodas feature square bases. Modern skyscrapers often use square plans for practical reasons.

Geometry also explores how squares interact with other shapes. You can fit a circle perfectly inside a square so that it touches the midpoint of each side. This is called an inscribed circle, and its radius is half the side length.

Conversely, a circumscribed circle can be drawn to pass through all four vertices of the square. The square is also the quadrilateral that provides the largest area for a given perimeter. This concept is part of the isoperimetric inequality. Furthermore, while squares are common in Euclidean geometry, they serve as the basis for "metric balls" in non-Euclidean systems like taxicab geometry.