Square

In Geometry, a square is a two-dimensional plane figure with four equal sides and all the four angles are equal to 90 degrees. The properties of rectangle are somewhat similar to a square, but the difference between the two is, a rectangle has only its opposite sides equal. Therefore, a rectangle is called a square only if all its four sides are of equal length.

Square Number of sides = 4 Number of vertices = 4 Area = Side2 Perimeter = 4(Side)

The other properties of the square such as area and perimeter also differ from that of a rectangle. Let us learn here in detail, what is a square and its properties along with solved examples.

Definition

Square is a regular quadrilateral, which has all the four sides of equal length and all four angles are also equal. The angles of the square are at right-angle or equal to 90-degrees. Also, the diagonals of the square are equal and bisect each other at 90 degrees.

A square can also be defined as a rectangle where two opposite sides have equal length.

The above figure represents a square where all the sides are equal and each angle equals 90 degrees.

Just like a rectangle, we can also consider a rhombus (which is also a convex quadrilateral and has all four sides equal), as a square, if it has a right vertex angle.

In the same way, a parallelogram with all its two adjacent equal sides and one right vertex angle is a square.

Shape of Square

A square is a four-sided polygon which has it’s all sides equal in length and the measure of the angles are 90 degrees. The shape of the square is such as, if it is cut by a plane from the center, then both the halves are symmetrical. Each half of the square then looks like a rectangle with opposite sides equal.

Properties of a Square

The most important properties of a square are listed below:

• All four interior angles are equal to 90°
• All four sides of the square are congruent or equal to each other
• The opposite sides of the square are parallel to each other
• The diagonals of the square bisect each other at 90°
• The two diagonals of the square are equal to each other
• The square has 4 vertices and 4 sides
• The diagonal of the square divide it into two similar isosceles triangles
• The length of diagonals is greater than the sides of the square

Area and Perimeter of Square

The area and perimeter are two main properties that define a square as a square. Let us learn them one by one:

Area

Area of the square is the region covered by it in a two-dimensional plane. The area here is equal to the square of the sides or side squared. It is measured in square unit.

           Area = side2 per square unit

If ‘a’ is the length of the side of square, then;

Area = a2 sq.unit

Perimeter

The perimeter of the square is equal to the sum of all its four sides. The unit of the perimeter remains the same as that of side-length of square.

Perimeter = Side + Side + Side + Side = 4 Side

           Perimeter = 4 × side of the square 

If ‘a’ is the length of side of square, then perimeter is:

Perimeter = 4a unit

Length of Diagonal of Square

The length of the diagonals of the square is equal to s√2, where s is the side of the square. As we know, the length of the diagonals is equal to each other. Therefore, by Pythagoras theorem, we can say, diagonal is the hypotenuse and the two sides of the triangle formed by diagonal of the square, are perpendicular and base.

Since, Hypotenuse2 = Base2 + Perpendicular2

Hence, Diagonal2 = Side2 + Side2

$\begin{array}{l}��������=\sqrt{2���{�}^2}\end{array}$

Hence, d = s√2

Where d is the length of the diagonal of a square and s is the side of the square.

Diagonal of square

Diagonal of square is a line segment that connects two opposite vertices of the square. As we have four vertices of a square, thus we can have two diagonals within a square. Diagonals of the square are always greater than its sides.

Below given are some important relation of diagonal of a square and other terms related to the square.

Relation between Diagonal ‘d’ and side ‘a’ of a square	$\begin{array}{l}�=�\sqrt{2}\end{array}$
Relation between Diagonal ‘d’ and Area ‘A’ of a Square-	$\begin{array}{l}�=\sqrt{2�}\end{array}$
Relation between Diagonal ‘d’ and Perimeter ‘P’ of a Square-	$\begin{array}{l}�=\frac{�}{2\sqrt{2}}\end{array}$
Relation between Diagonal ‘d’ and Circumradius ‘R’ of a square:	d = 2R
Relation between Diagonal ‘d’ and diameter of the Circumcircle	$\begin{array}{l}�={�}_{�}\end{array}$
Relation between Diagonal ‘d’ and In-radius (r) of a circle-	$\begin{array}{l}�=2\sqrt{2}�\end{array}$
Relation between Diagonal ‘d’ and diameter of the In-circle	$\begin{array}{l}�=\sqrt{2}{�}_{�}\end{array}$
Relation between diagonal and length of the segment l-	$\begin{array}{l}�=�\frac{2\sqrt{10}}{5}\end{array}$

Solved Examples

Problem 1: Let a square have side equal to 6 cm. Find out its area, perimeter and length of diagonal.

Solution: Given, side of the square, s = 6 cm

Area of the square = s2 = 62 = 36 cm2

Perimeter of the square = 4 ×  s = 4 × 6 cm = 24cm

Length of the diagonal of square = s√2 = 6 × 1.414 = 8.484

Problem 2: If the area of the square is 16 sq.cm., then what is the length of its sides. Also find the perimeter of square.

Solution: Given, Area of square = 16 sq.cm.

As we know,

area of square =side2

Therefore, by substituting the value of area, we get;

16 = side2

side = √16 = √(4×4) = 4 cm

Hence, the length of the side of square is 4 cm.

Now, the perimeter of square is:

P = 4 x side = 4 x 4 = 16 cm.