Properties of Squares

A square is a rectangle with four equal sides. Although relatively simple and straightforward to deal with, squares have several interesting and notable properties.

The fundamental definition of a square is as follows:

A square is a quadrilateral whose interior angles and side lengths are all equal.

A square is both a rectangle and a rhombus and inherits the properties of both (except with both sides equal to each other). The square is the area-maximizing rectangle.

Property 1. Each of the interior angles of a square is \( 90^\circ \).

Property 2. The diagonals of a square bisect each other.

Property 3. The opposite sides of a square are parallel.

Property 4. A square whose side length is \( s \) has area \( s^2 \).

Property 5. A square whose side length is \( s \) has perimeter \( 4s \).

Property 6. A square whose side length is \( s \) has a diagonal of length \( s\sqrt{2} \).

Property 7. The diagonals of a square are equal.

Property 7. Let \( O \) be the intersection of the diagonals of a square. There exists a circumcircle centered at \( O \) whose radius is equal to half of the length of a diagonal.

Property 8. Each diagonal of a square is a diameter of its circumcircle.

Additionally, for a square one can show that the diagonals are perpendicular bisectors.

Property 9. The diagonals of a square are perpendicular bisectors.

The four triangles bounded by the perimeter of the square and the diagonals are congruent by SSS. Therefore, the four central angles formed at the intersection of the diagonals must be equal, each measuring \( \frac{360^\circ}4 = 90^\circ \).

Alternatively, one can simply argue that the angles must be right angles by symmetry.

However, while a rectangle that is not a square does not have an incircle, all squares have incircles. There exists a point, the center of the square, that is both equidistant from all four sides and all four vertices.

Property 10. Let \( O \) be the intersection of the diagonals of a square. There exists an incircle centered at \( O \) whose radius is equal to half the length of a side.

Square inside a square

Suppose a square is inscribed inside the incircle of a larger square of side length \( S \). Find the side length \( s \) of the inscribed square, and determine the ratio of the area of the inscribed square to that of the larger square.

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The diameter of the incircle of the larger square is equal to \( S\). At the same time, the incircle of the larger square is also the circumcircle of the smaller square, which must have a diagonal equal to the diameter of the circumcircle. Therefore, \( S = s \sqrt{2} \), or \( s = \frac{S}{\sqrt{2}} \).

It follows that the ratio of areas is \[ \frac{s^2}{S^2} = \frac{\ \ \dfrac{S^2}{2}\ \ }{S^2} = \frac12.\ _\square \]

Area bounded by an arc and square

A square with side length \( s \) is circumscribed, as shown. Determine the area of the shaded area. (Note this this is a special case of the analogous problem in the properties of rectangles article.)

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We can consider the shaded area as equal to the area inside the arc that subtends the shaded area minus the fourth of the square (a triangular wedge) that is under the arc but not part of the shaded area.

The arc that bounds the shaded area is subtended by an angle of \( 90^\circ \), or one-fourth of the circle Therefore, the area under the arc is \( \frac{\pi R^2}4 = \frac{\pi s^2}8 \), where \( R = \frac{s \sqrt{2}}2 \) is the radius of the circle. Finally, subtracting a fourth of the square's area gives a total shaded area of \( \frac{s^2}{4} \left(\frac{\pi}{2} - 1 \right) \). \(_\square\)