Square

A square is a kind of quadrilateral. In fact, a square is a type of rectangle in which all four sides are equal. All squares are rectangles, but not all rectangles are squares. There are many square objects around you. Each square shape is characterized by only one dimension, its side length. In this chapter, we will learn about the concept of square, its shape, important formulas like the area and perimeter, and properties.

Table of Contents

What is Square?

A square is a closed two-dimensional figure with four sides and four corners. The length of all four sides is equal and parallel to each other. The basic figure of a square is shown below.

A square is a quadrilateral in which:

  • the opposite sides are parallel.
  • all four sides are equal.
  • all angles measure 90°.

Shape of Square

Diagonal of a Square

The diagonal of a square is a line segment that joins any two of its opposite 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 triangles so formed.

Diagonals of a Square

Let s be the side length and d be the diagonal length of a square. We can use the Pythagoras theorem for the triangle ABD: d2 = s2 + s2

Taking square root on both sides gives, √(d2) = √( 2s2). Thus, the diagonal of a square formula is: Diagonal of Square (d) = √2 × s

Formulas of Square

We know that a square is a four-sided figure with equal sides. There are two formulas that apply to a square. The first one is to calculate its area, the second is to calculate its perimeter. Let's learn both the formulas in detail.

Area of a Square 

The area of a square is the space occupied by the square. 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 with side 's' is ssquare units. When the diagonal d of the square is known, the formula for finding the area of the square is: (d2 / 2) square units

Perimeter of a square

The perimeter of any shape is the length present outside of the shape. The perimeter 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) units

Properties of a Square

A square is a closed figure of four equal sides and the angle formed by adjacent sides is 90 degrees. 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.

  • The length of the diagonal with sides s is √2 × s

  • 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.

How to Construct a Square?

Here is a step by step process for you to construct a square using a ruler and a compass. These are the following steps to construct a square:

  • Step 1: Draw a line segment PQ of any dimension.
  • Step 2: Extend the same line PQ. Use a compass to draw two arcs on each side of Q and name the points as U and V.
  • Step 3: Using the compass from point U and V, draw arcs above the point Q. The point where the arcs meet is named W.
  • Step 4: Now, draw a line from Q to W.
  • Step 5: Set the compass to any radius. Draw an arc from point P above the same point. Now move the compass with the same radius to point Q and draw an arc across QW and named it R. This is the vertex of the square.
  • Step 6: Using the compass with the same radius, draw an arc from point R on the previously drawn arc above point P. The point of intersection is named S.
  • Step 7: Join the points R and S as well as P and S to get the square PQRS.  

constructing square stepwise

  • Step 8: Erase all lines and arcs except the ones made in the last two steps. The final figure is that of a square.

construction of square stepwise 1


FAQs on Square

Is Square a Polygon?

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

Can a Rhombus be a Square?

A square can be a rhombus but a rhombus can not be a square because all interior angles of the square are equal to 90º but all interior angles of a rhombus are not necessarily equal to 90º.

How can you Identify a Square shape?

A square can be identified as a polygon that consists of four equal sides and all the interior angles as 90º.

Is Square a Regular Polygon?

A regular polygon is a polygon whose length is equal on all sides with equilateral angles. Since a square has all its sides equal, it is 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 are parallel, and 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 the square. The area of a square is the square of its side length. The formula for calculating the area of a square is: Area =  (side)2.

Can you List Some Properties of Square?

The basic properties of a square are:

  • All sides are parallel and equal.
  • The interior angle of a square at each vertex is 90°.
  • The diagonals are equal and they bisect each other at 90°.

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

No, the area of the square is not necessarily the same as the area of the rectangle because every square is a rectangle with length and breadth equal but all rectangles are not square. The formula to calculate the area of a square is (side)2 and that of a rectangle is Length × Breadth.

What is the Formula for Perimeter of a Square?

The perimeter of a square is four times the side of the square. Perimeter = 4 (Side) units. Perimeter is the total of the lengths of the boundaries of the square. 


Related Topics to Square

Given below is the list of topics that are closely connected to square. These topics will also give you a glimpse of how such concepts are covered in Cuemath.


Solved Examples

Example 1: Peter ordered a square photo frame of side 6 inches. Can you find its area?

Solution:

We know that the area of a square of side s is s2. Thus, s2 = (6 × 6) square inches = 36 square inches. Therefore, the area of the photo frame is 36 square inches.

Example 2: The length of each side of a park is 400 yards. Find the perimeter of the park.

Solution:

The length of each side of the park is 400 yards. Thus,  (4 × Side) units = (4 × 400) yards = 1600 yards. Therefore, the perimeter of the park is 1600 yards.

Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

 

  
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