Perimeter and area

The area of a square is the product of the length of its sides.

The perimeter of a square whose four sides have length ${\displaystyle \ell }$ is

${\displaystyle P=4\ell }$

and the area A is

${\displaystyle A=\ell ^{2}.}$^[1]

Since four squared equals sixteen, a four by four square has an area equal to its perimeter. The only other quadrilateral with such a property is that of a three by six rectangle.

In classical times, the second power was described in terms of the area of a square, as in the above formula. This led to the use of the term square to mean raising to the second power.

The area can also be calculated using the diagonal d according to

${\displaystyle A={\frac {d^{2}}{2}}.}$

In terms of the circumradius R, the area of a square is

${\displaystyle A=2R^{2};}$

since the area of the circle is ${\displaystyle \pi R^{2},}$ the square fills ${\displaystyle 2/\pi \approx 0.6366}$ of its circumscribed circle.

In terms of the inradius r, the area of the square is

${\displaystyle A=4r^{2};}$

hence the area of the inscribed circle is ${\displaystyle \pi /4\approx 0.7854}$ of that of the square.

Because it is a regular polygon, a square is the quadrilateral of least perimeter enclosing a given area. Dually, a square is the quadrilateral containing the largest area within a given perimeter.^[6] Indeed, if A and P are the area and perimeter enclosed by a quadrilateral, then the following isoperimetric inequality holds:

${\displaystyle 16A\leq P^{2}}$

with equality if and only if the quadrilateral is a square.

Other facts

• The diagonals of a square are ${\displaystyle {\sqrt {2}}}$ (about 1.414) times the length of a side of the square. This value, known as the square root of 2 or Pythagoras' constant,^[1] was the first number proven to be irrational.
• A square can also be defined as a parallelogram with equal diagonals that bisect the angles.
• If a figure is both a rectangle (right angles) and a rhombus (equal edge lengths), then it is a square.
• A square has a larger area than any other quadrilateral with the same perimeter.^[7]
• A square tiling is one of three regular tilings of the plane (the others are the equilateral triangle and the regular hexagon).
• The square is in two families of polytopes in two dimensions: hypercube and the cross-polytope. The Schläfli symbol for the square is {4}.
• The square is a highly symmetric object. There are four lines of reflectional symmetry and it has rotational symmetry of order 4 (through 90°, 180° and 270°). Its symmetry group is the dihedral group D₄.
• A square can be inscribed inside any regular polygon. The only other polygon with this property is the equilateral triangle.
• If the inscribed circle of a square ABCD has tangency points E on AB, F on BC, G on CD, and H on DA, then for any point P on the inscribed circle,^[8]

${\displaystyle 2(PH^{2}-PE^{2})=PD^{2}-PB^{2}.}$

• If ${\displaystyle d_{i}}$ is the distance from an arbitrary point in the plane to the i-th vertex of a square and ${\displaystyle R}$ is the circumradius of the square, then^[9]

${\displaystyle {\frac {d_{1}^{4}+d_{2}^{4}+d_{3}^{4}+d_{4}^{4}}{4}}+3R^{4}=\left({\frac {d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}}{4}}+R^{2}\right)^{2}.}$

• If ${\displaystyle L}$ and ${\displaystyle d_{i}}$ are the distances from an arbitrary point in the plane to the centroid of the square and its four vertices respectively, then ^[10]

${\displaystyle d_{1}^{2}+d_{3}^{2}=d_{2}^{2}+d_{4}^{2}=2(R^{2}+L^{2})}$

and

${\displaystyle d_{1}^{2}d_{3}^{2}+d_{2}^{2}d_{4}^{2}=2(R^{4}+L^{4}),}$

where ${\displaystyle R}$ is the circumradius of the square.