Square

A square is a quadrilateral with four equal sides. There are many objects around us that are in the shape of a square. Each square shape is identified by its equal sides and its interior angles that are equal to 90°. Let us learn more about the properties of a square, its formulas, and its construction.

A square is a closed two-dimensional shape (2D shape) with four sides. All four sides of a square are equal and parallel to each other. The basic figure of a square is shown below.

Square Definition

A square is a quadrilateral in which:

• The opposite sides are parallel.
• All four sides are equal.
• All angles measure 90°.

A square is a closed figure of four equal sides and the interior angles of a square are equal to 90°. A square can have a wide range of properties. Some of the important properties of a square are given below.

• A square is a quadrilateral with 4 sides and 4 vertices.
• All four sides of the square are equal to each other.
• The opposite sides of a square are parallel to each other.
• The interior angle of a square at each vertex is 90°.
• The sum of all interior angles is 360°.
• The diagonals of a square bisect each other at 90°.
• The length of the diagonals is equal.
• Since the sides of a square are parallel, it is also called a parallelogram.
• The length of the diagonals in a square is greater than its sides.
• The diagonals divide the square into two congruent triangles.

There are some properties that are common to a square and a rectangle. The following points show all the common properties that define a rectangle and a square.

• A square and a rectangle are quadrilaterals with 4 sides and 4 vertices.
• The opposite sides of a square and a rectangle are parallel to each other.
• Each interior angle of a square and a rectangle is 90°.
• The sum of all interior angles of a square and a rectangle is 360°.
• The diagonal of a square and a rectangle divides them into 2 right-angled triangles.
• Since the opposite sides of a square and a rectangle are parallel, they are also called parallelograms.

We know that a square is a four-sided figure with equal sides. There are three basic square formulas that are commonly used in geometry. The first one is to calculate its area, the second is to calculate its perimeter and the third is the diagonal of a square formula. Let us learn these square formulas in detail.

Area of a Square

The area of a square is the space occupied by it. Some examples of square shapes are chessboard, square wall clock, etc. We can use the formula of the area of a square to find the space occupied by these objects. The formula for the area of a square is expressed as, Area of square = s²; where 's' is the side of the square. It is expressed in square units like cm², m², and so on.

Perimeter of a Square

The perimeter of a square is the total length of its boundary. Therefore, the perimeter of a square can be calculated by adding the length of all the sides. Since a square has four sides, we must add all the four sides of a square to find its perimeter.

We can use the formula of the perimeter of a square to find the length of its boundary. Perimeter of a square = side + side + side + side. Therefore, Perimeter of Square = (4 × Side). It is expressed in linear units like cm, m, inches, and so on.

Diagonal of a Square Formula

The diagonal of a square is a line segment that joins any two of its non-adjacent vertices. In the following square, AC and BD are the diagonals of the square. Observe that the lengths of the lines AC and BD are the same. A diagonal cuts a square into two equal right triangles and each diagonal forms the hypotenuse of the right-angled triangles so formed.

Let us see how the formula for the diagonal of a square is derived. Following the square given above, let 'a' be the side length and 'd' be the diagonal length of a square. We can use the Pythagoras theorem for the triangle ADC: d² = a² + a²

Taking square root on both sides gives, √(d²) = √( 2a²). Thus, the diagonal of a square formula is: Diagonal of Square (d) = √2 × a

The basic construction of a square can be done using a ruler and a compass. The following steps show how to construct a square. For example, if we need to draw a square in which all the sides are of 6 cm.

• Step 1: Draw a line segment PQ of 6 cm.
• Step 2: Extend the line segment PQ in one direction, say to the right. Now, take the compass and keeping Q as the center draw one arc on line segment PQ and another arc of the same length on the extended side as shown in the figure. Mark the points as U and V.
• Step 3: Keep the compass on points U and V and draw arcs above point Q such that they intersect each other. The point where the arcs meet is named W.
• Step 4: Now, draw a line from Q to W. This is a perpendicular that is drawn on the line segment PQ.
• Step 5: Set the compass such that it measures 6 cm and draw an arc from point Q across QW and named it R. This is one vertex of the square.
• Step 6: Using the compass with the same length of 6 cm, draw an arc from point P and draw an arc above. Keeping the same length, draw another arc keeping R as the center such that it intersects the arc created from point P. Mark this intersecting point as S.
• Step 7: Join the points R and S and then P and S to get the square PQRS.
• Step 8: PQRS is a square in which all the sides are 6 cm and all the angles are 90°.

☛ Related Articles

• Square Pyramid
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What is a Square in Geometry?

A square is a four-sided regular polygon which is also known as a quadrilateral with four equal sides. It has four equal angles that measure 90° each. A square, in geometry, can also be defined as a parallelogram because it has two opposite sides that are parallel to each other.

What are the Properties of a Square?

The basic properties of a square are listed below:

• All four interior angles of a square are equal and each measures 90°.
• All four sides of the square are equal to each other.
• The opposite sides of a square are parallel to each other.
• A square is also considered to be a rectangle because all its angles measure 90° and its opposite sides are equal and parallel.

What are the Properties of the Diagonal of a Square?

The properties of a square related to the diagonal are listed below:

• The diagonals of a square are equal in length.
• The diagonals of a square bisect each other.
• The diagonals of a square are perpendicular to each other.

Is Square a Polygon?

Yes, a square is a polygon because it is a closed shape that consists of four sides and four vertices and a polygon is a closed shape joined end-to-end with straight lines.

Can a Rhombus be a Square?

No, a rhombus cannot be a square because all interior angles of the square are equal to 90º but all interior angles of a rhombus may not necessarily be equal to 90º. However, a square can be a rhombus because all the sides in a rhombus are of equal length and a square fulfills this property.

How to Identify a Square shape?

A square can be identified as a polygon that consists of four equal sides and all the interior angles are 90º. This means if a polygon has four equal sides and all its interior angles are 90°, it can be identified as a square.

Is Square a Regular Polygon?

Yes, a square is a regular polygon because a regular polygon is a polygon in which all the sides are of equal length and all the angles are of equal measure. Since a square fulfills this property, it is considered to be a regular polygon.

Is Square a Trapezoid?

Yes, a square is a trapezoid because a trapezoid is a quadrilateral in which one pair of opposite sides is parallel, and a square fulfills this property because in a square both pairs of opposite sides are parallel.

What is the Formula for the Area of a Square?

The area of a square is the space occupied by it. The formula for calculating the area of a square is: Area = (side)².

Is the Area of the Square Equal to the Area of the Rectangle?

No, the area of a square is not equal to the area of the rectangle. In geometry, according to the properties of a square and rectangle we consider every square to be a rectangle, but all rectangles are not squares. The formula to calculate the area of a square = (side)² and the area of a rectangle = Length × Width.

What is the Formula for the Perimeter of a Square?

The perimeter of a square is the total length of its boundary. The formula for the perimeter of a square is, Perimeter of a square = 4 ×(Side) and it is expressed in linear units like cm, inches, and so on.