Area of a square is one of the most important aspects associated with the chapter mensuration. Square is a two-dimensional figure with equal sides and angles. The area of a square gives the amount of space occupied by the square in it. The area of a square is generally denoted by using square units like square meters, square centimetres etc. we can find the square’s area using the formula \({\rm{side}} \times {\rm{side}}.\)

What is a Square?

A square is a closed two-dimensional figure. It has four equal sides and four equal angles. The four equal sides of the square form the four right angles at the vertices. The sum of the total length of four sides of a square is known as its perimeter. The total space occupied by the square is called the area of the square. 

A square is a special type of quadrilateral. It is also called a regular quadrilateral as it has all equal sides and equal angles. The sum of all four angles of the square is \({360^{\rm{o}}}.\) Moreover, a square is a special type of rectangle, in which whose length and breadth are equal in measures.

A square is a quadrilateral, which has special properties as mentioned below:

1. All sides are of equal length.
2. All angles are equal, and each measures a right angle.
3. Diagonals are perpendicular bisectors.

Properties of Square

A square is a two-dimensional figure in which all sides are of equal length. Some specific properties of the square, which differentiate from all other quadrilaterals, are mentioned below:

1. A square has all four sides of equal length.

2. In a square, all angles are equal, and each being a right angle.

3. In a square, opposite sides are parallel.

4. The sum of all angles in a square is \(360\) degrees.

5. In a square, diagonals bisect each other, and they are perpendicular.

Area of Square: Definition

In geometry, the area is space enclosed by an object. The area of a square is the surface or space occupied by the square in it. The area of the square is found by using the formula: \({\rm{side \times side = sid}}{{\rm{e}}^{\rm{2}}}.\)

Thus, the area of a square is the product of any two sides of the square. It is also defined as the square of the sides. The unit of area of a square is given in square units like square meters, square centimetres, square feet, etc., as the area of the square is the product of the two sides.

We can find the area of the square by using the diameter and perimeter of the square also. The area of the square is calculated by using the number of squares in it and taking each square area as \(1\) square unit.

Example:
Area of the square with side \(5\) units found as follows:

Dividing the given square into an equal number of squares as shown below, there are a total of \(25\) small squares. Therefore, the area of the square is \(25\) square units.

Area of Square: Formula

The area of the square is found by using the formula: \({\rm{side \times side = sid}}{{\rm{e}}^{\rm{2}}}.\)

The unit of the area of the square is given in square units like square meters, square centimetres, square feet, etc., as the area of the square is the product of the two sides.

The area of the square with side is \(“a”\) units is given by \(a \times a = {a^2}.\) Algebraically, the area of the square can be calculated by squaring the number representing the measure of the side of the given square.

Example:
The area of a square with side measures \(5\) units is obtained by squaring the given number of the measurement of the side of the square. Area\( = {5^2} = 25\,{\rm{sq}}{\rm{.}}\,{\rm{units}}.\)

Area of Square by Diagonal

We know that in a square, diagonals are equal and perpendicular to each other. A diagonal divides the square into two congruent right triangles (each angle in a square is \({90^{\rm{o}}}.\))

Consider a square with side \(“a”\) units and diagonal \(“d”\) as shown below:

In a right triangle, \(ABC,\)

By Pythagoras theorem:
\({{\rm{( hypotenuse )}}^{\rm{2}}}{\rm{ = (side}}{{\rm{)}}^{\rm{2}}}{\rm{ + (side}}{{\rm{)}}^{\rm{2}}}\)
\( \Rightarrow A{C^2} = A{B^2} + B{C^2}\)
\( \Rightarrow {d^2} = {a^2} + {a^2}\)
\( \Rightarrow {d^2} = 2{a^2}\)

Applying square on both sides of the above equation,
\( \Rightarrow d = \sqrt 2 a\)
\( \Rightarrow a = \frac{d}{{\sqrt 2 }}\)

So, the side of the square in terms of the diagonal is given by \({\rm{side}} = \frac{{{\rm{ diagonal }}}}{{\sqrt 2 }}.\)

We know that area of the square is \({\rm{sid}}{{\rm{e}}^{\rm{2}}}.\)
So, the area of the square is \({\left( {\frac{{{\rm{ diagonal }}}}{{\sqrt 2 }}} \right)^2} = \frac{{{\rm{ diagonal}}{{\rm{ }}^2}}}{2}\)
Therefore, the area of the square can be found by using the diagonal by using the below formula:

\({\rm{Area}}\,{\rm{of}}\,{\rm{the}}\,{\rm{square}} = \frac{{{\rm{ diagonal}}{{\rm{ }}^2}}}{2}\)

Area of Square Using Perimeter

We know that sum of lengths of all sides of the square is called the perimeter of the square. So, the perimeter of the square is \(4×\)side.

The perimeter of the square with side \(“a”\) is \(4a.\)
So, from the above, we can write, 
\(a = \frac{{{\rm{ perimeter }}}}{4}.\)

We know that area of the square is
\({\rm{sid}}{{\rm{e}}^{\rm{2}}} = {\left( {\frac{{{\rm{ perimeter }}}}{4}} \right)^2} = \frac{{{\rm{ perimeter}}{{\rm{ }}^2}}}{{16}}.\)
Therefore, the area of the square can be found by using the perimeter as given below:

\({\rm{Area}}\,{\rm{of}}\,{\rm{the}}\,{\rm{square}} = \frac{{{\rm{ perimeter}}{{\rm{ }}^2}}}{{16}}\)

Area of a Square Pyramid

The square pyramid is the pyramid having the square as the base and isosceles triangles as the lateral faces. Thus, it has four lateral surfaces (isosceles triangles) and a base (square). All these isosceles triangles are congruent. Each of its sides coincides with the side of the base (square).

We know that “surface” means ” the exterior or outside portion of an object or body”. Thus, the total surface area of a square pyramid is equal to the sum of its lateral surfaces (triangles) and base (square). 

The surface area of the square pyramid is equal to the sum of the areas of the base (square) and all four lateral faces (isosceles triangle).

\({\rm{Area}} = {a^2} + 4 \times {\rm{area}}\,{\rm{of}}\,{\rm{an}}\,{\rm{isosceles}}\,{\rm{triangle}}\,{\rm{with}}\,{\rm{sides}}\,a,l\,{\rm{and}}\,l\)
\( = {a^2} + 4 \times \frac{1}{2} \times a \times l\)
\( = {a^2} + 2al\)

Area of Square Formulas

The area of the square with side \(“a”,\) diagonal \(“d”\) and perimeter \(“P”\) is given below:

Parameter	Formula
Area of the square using side \((a)\)	\({a^2}\)
Area of the square using diagonal \((d)\)	\(\frac{{{d^2}}}{2}\)
Area of the square using the perimeter \((P)\)	\(\frac{{{p^2}}}{{16}}\)

Solved Examples – Area of Square Formula

Q.1. Find the area of the square whose sides measure \({\rm{6}}\,{\rm{cm}}\) each.
Ans: Given side of the square is \({\rm{6}}\,{\rm{cm}}.\)

We know the side×side gives that area of the square.
\({\rm{Area}} = 6 \times 6 = {6^2} = 36\;{\rm{c}}{{\rm{m}}^2}\)
Hence, the area of the given square is \(36\;{\rm{c}}{{\rm{m}}^2}.\)

Q.2. A swimming pool has an area of \({\rm{100}}\,{{\rm{m}}^{\rm{2}}}{\rm{,}}\) which is in the shape of a square. Find the length of the side of the swimming pool.
Ans: Given a swimming pool is in the shape of a square.
Given, area of the swimming pool (square) is \(100\,{{\rm{m}}^2}.\)
We know that area of the square is \({\rm{sid}}{{\rm{e}}^{\rm{2}}}.\)
\( \Rightarrow {\rm{sid}}{{\rm{e}}^{\rm{2}}}{\rm{ = 100}}\,{{\rm{m}}^{\rm{2}}}\)
Applying square root on both sides of the equation
\( \Rightarrow {\rm{side = }}\sqrt {{\rm{100}}} \)
\( \Rightarrow {\rm{side = 10\;m}}\)
Hence, the length of the side of the swimming pool is \({\rm{10\;m}}.\)

Q.3. Find the area of the square carpet whose length of the diagonal is \(8\) feet.
Ans: Given the length of the diagonal of the square carpet \((d) = 8\) feet.
We know that area of the square with diagonal \((d)\) is given by \(\frac{{{d^2}}}{2}.\)
\( \Rightarrow \frac{{{8^2}}}{2} = \frac{{64}}{2} = 32\) square feet.
Therefore, the area of the given square carpet is \(32\) square feet.

Q.4. The perimeter of the square-shaped carrom board is given as \({\rm{240}}\,{\rm{cm}}{\rm{.}}\) Find the area of the carrom board?
Ans: Given perimeter of the square-shaped carrom board is \({\rm{240}}\,{\rm{cm}}{\rm{.}}\)
We know that area of the square, whose perimeter is given as \(\frac{{{\rm{ perimeter}}{{\rm{ }}^2}}}{{16}}.\)
\( \Rightarrow \frac{{{{240}^2}}}{{16}}\)
\( \Rightarrow \frac{{57600}}{{16}} = 3600\,{\rm{c}}{{\rm{m}}^2}\)
Hence, the area of the given carrom board is \(3600\,{\rm{c}}{{\rm{m}}^2}.\)

Q.5. Find the area of the base of the square pyramid, whose side is \({\rm{12}}\,{\rm{cm}}{\rm{.}}\)
Ans: We know that base of the square pyramid is in the shape of a square.

Given side of the base of the square pyramid is \({\rm{12}}\,{\rm{cm}}{\rm{.}}\)
We know that area of the square is \({\rm{sid}}{{\rm{e}}^{\rm{2}}}\)
\( \Rightarrow {12^2} = 144\;{\rm{c}}{{\rm{m}}^2}\)
Hence, the area of the base of the square pyramid is \(144\;{\rm{c}}{{\rm{m}}^2}.\)

Summary

In this article, we have studied the area of the square formula, which is used to calculate the area of the square-shaped objects. We have studied the definition of the area of the square, which measures the amount of surface or space in it.

We have discussed the definition of a square which is a two-dimensional figure with all sides are equal, and all angles are equal. We also studied the properties of the square. We have studied how to find the area of the squares using side, diagonal and perimeter. We also discussed how to find the area of the square by using different formulas with the help of solved examples.

Frequently Asked Questions (FAQs) – Area of Square

Frequently asked questions related to area of square are listed as follows:

Q.1. What is the area of the square formula?
Ans: The area of the square formula is the formula used to find the amount of surface or space in the square.

Q.2. What is the formula for finding the area of the square?
Ans: The area of the square can be calculated by using the side, perimeter and diagonal of the square. The general formula used to calculate the area of the square is \({\rm{side}} \times {\rm{side}}.\)

Q.3. What is the area of the square, which is inscribed in the circle?
Ans: We know that if a square is inscribed in the circle, then the diameters of the circle and the square are equal.
The area of the square whose diagonal is known is given by \(\frac{{{\rm{ diagona}}{{\rm{l}}^2}}}{2}\)

Q.4. How do we find the area of the square by using the perimeter?
Ans: The area of the square is found by using the perimeter also. The formula for the area of the square by using the perimeter is given below:
\(\frac{{{\rm{ perimete}}{{\rm{r}}^2}}}{{16}}.\)

Q.5. What is the formula for the area of the square pyramid?

Ans: The area of the square pyramid with the length of the base \(” a “\) and slant height \(” l “\) is given by \({a^2} + 2al.\)

Q.6. What are the units used for the area of the square?
Ans: The area of the square is the product of any two sides. So, the area of the square is found using the square units, such as square meters, square centimetres, square feet, etc.

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