Definition:Quadrilateral/Square

Definition

A square is a regular quadrilateral.

That is, a regular polygon with $4$ sides.

That is, a square is a plane figure with four sides all the same length and whose angles are all equal.

Euclid's Definition

In the words of Euclid:

Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.

(The Elements: Book $\text{I}$: Definition $22$)

Also known as

Writers of popular mathematical literature, appealing to the non-mathematically inclined, can be seen using the term perfect square.

However, as there is no such thing as an imperfect square, this sort of usage is not welcome on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also see

• Internal Angles of Square: the angles of a square are all right angles

• Area of Square

• Results about squares in the context of geometry can be found here.

Sources

• 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.22$: Corollary $3$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $4$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): parallelogram
• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.4$: Euclid (flourished ca. $300$ B.C.)
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $4$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): square: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): square: 1.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): square