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.
Contents
Basic 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.
Points ABCD are midpoints of the sides of the larger square. If the larger square has area 60, what's the small square's area?
A square (the geometric figure) is divided into 9 identical smaller squares, like a tic-tac-toe board. If the original square has a side length of 3 (and thus the 9 small squares all have a side length of 1), and you remove the central small square, what is the area of the remaining figure?
Consider a square \( ABCD \) with side length 2. Let \(E\) be the midpoint of \(AB\), \(F\) the midpoint of \(BC\), and \(P\) and \(Q\) the points at which line segment \(\overline{AF}\) intersects \(\overline{DE}\) and \(\overline{DB}\), respectively.
What is the area of \( EBQP? \)
Additional Properties
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.
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.)
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 \( \frac{s^2}{4} \left(\frac{\pi}{2} - 1 \right) \). \(_\square\)
In a large square, the incircle is drawn (with diameter equal to the side length of the large square).
In the circle, a smaller square is inscribed.
What is the ratio of the area of the smaller square to the area of the larger square?
The diagram above shows a large square, whose midpoints are connected up to form a smaller square. We then connect up the midpoints of the smaller square, to obtain the inner shaded square.
What fraction of the large square is shaded?
Note: Give your answer as a decimal to 2 decimal places.
In the figure above, we have a square and a circle inside a larger square.
Find the radius of the circle, to 3 decimal places.
The ratio of the area of the square inscribed in a semicircle to the area of the square inscribed in the entire circle is \(\text{__________}.\)
\(\)
Details and Assumptions:
- If your answer is 10:11, then write it as 1011.
- Note that the ratio remains the same in all cases.
A chord of a circle divides the circle into two parts such that the squares inscribed in the two parts have areas 16 and 144, respectively.
The radius of the circle is \(\text{__________}.\)