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Area of Square – Definition, Formula & Examples

area of square

The area of a 2D shape (plane figure) can be viewed as the number of square units needed to fill a square. In general, the area is defined as the region occupied inside the boundary of a flat object or 2D figure. The measurement of the area is done in square units such as $cm^{2}$, $m^{2}$, $ft^{2}$, etc.

For the computation of area, there are pre-defined formulas for squares, rectangles, circles, triangles, etc. In this article, let’s learn about the area of a square.

Square – A 2D Plane Figure

A square, in geometry, is a plane 2D shape with four equal sides and four right angles $\left(90^{\circ} \right)$. A square is a special kind of rectangle (an equilateral one) and a special kind of parallelogram (an equilateral and equiangular one).

area of square

What is the Area of Square?

The area of a square is the measure of the space or surface occupied by it. Let’s consider a square of length $6$ units, i.e., a square having length of all of its edges (sides) of $6$ units.

Let’s further divide this square into a number of small squares of edge $1$ unit.

area of square

How many such small squares are there inside the big square?

There are $36$ small squares of edge $1$ unit each inside the bigger square. So, the area of the bigger square is $36 unit^{2}$.

Note: The unit can be $mm$, $cm$, $m$, $ft$, $in$, etc.

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Area of a Square Formula Using Sides

The formula to find the area of a square of the length of each side (or edge) is given by $A = s^{2}$, where $s$ is the length of each side of a square.

For example, the area of a square of side $3 cm$ is $3^{2} = 9 cm^{2}$.

How to Calculate Area of Square Using Sides?

The area of a square is equal to the square of its length. The following steps are used to find the area of a square:

Step 1: Note down the length of a square

Step 2: Substitute the value of the length of a square in the formula

Step 4: Simplify the expression in the formula to get the area in square unit

Examples

Ex 1: Find the area of a square whose side is $17 mm$.

Length of side of square $s = 17 mm$

Area of a square $A = s^{2}$

Substitute the value of $s$ in the formula.

$A = 17^{2} = 289 mm^{2}$

Therefore, the area of a square of side $17 mm$ is $289 mm^{2}$.

Ex 2: Find the area of a square whose side is $1.25 m$

Length of side of square $s = 1.25 m$

Area of a square $A = s^{2}$

Substitute the value of $s$ in the formula.

$A = 1.25^{2} = 1.5625 m^{2}$

Therefore, the area of a square of side $1.25 m$ is $1.5625 m^{2}$.

Ex 3: Find the length of the side of a square whose area is $1225 in^{2}$.

Area of square = $1225 in^{2}$

Let the length of the side of a square be $s in$

Therefore, $s^{2} = 1225$

Taking the square root of both sides

$s = \sqrt{1225} = 35 in$

Therefore, the length of a side of a square of area $1225 in^{2}$ is $35 in$.

Ex 4: A square courtyard of length $5 m$ is to be covered by square tiles of length $25 cm$. How many tiles will be required to cover the courtyard?

Length of the side of a square courtyard = $S = 5 m = 5 \times 100 = 500 cm$

Length of the side of a square tile = $s = 25 cm$

Area of courtyard = $S^{2} = 500^{2} = 250000 cm^{2}$

Area of one tile = $s^{2} = 25^{2} = 625 cm^{2}$

Number of tiles = $\frac {250000}{625} = 400$

Therefore, to cover a square courtyard of length $5 m$, the number of square tiles of length $25 cm$ needed is $400$.

Alternatively, you can convert the dimensions of a tile into metre and calculate the number of tiles required.

Length of the side of a square courtyard = $S = 5 m$

Length of the side of a square tile = $s = 25 cm = \frac {25}{100} = 0.25 m$

Area of courtyard = $S^{2} = 5^{2} = 25 m^{2}$

Area of one tile = $s^{2} = 0.25^{2} = 0.0625 cm^{2}$

Number of tiles = $\frac {25}{0.0625} = 400$

Therefore, to cover a square courtyard of length $5 m$, the number of square tiles of length $25 cm$ needed is $400$.

10 Famous Mathematicians

Area of a Square Formula Using a Diagonal

You can also find the area of a square if the length of its diagonal is known. In this case, you use the Pythagoras Theorem to find the area of a square.

Let’s consider a square of the length of side $s$, and diagonal $d$.

area of square

By Pythagoras Theorem, we get $d^{2} = s^{2} + s^{2} => d^{2} = 2s^{2} =>s = \frac {d}{\sqrt{2}}$

Therefore, the area of a square is given by $A = s^{2} = \left({\frac {d}{\sqrt{2}}}\right)^2 => A = \frac {d^{2}}{2}$.

You can also find the area of a square if the length of its diagonal is known. In this case, you use the Pythagoras Theorem to find the area of a square.

Let’s consider a square of the length of side $s$, and diagonal $d$.

By Pythagoras Theorem, we get $d^{2} = s^{2} + s^{2} => d^{2} = 2s^{2} =>s = \frac {d}{\sqrt{2}}$

Therefore, the area of a square is given by $A = s^{2} = \left({\frac {d}{\sqrt{2}}}\right)^2 => A = \frac {d^{2}}{2}$.

Note: Diagonals of a square are equal

Examples

Ex 1: Find the area of a square whose diagonal is $8 cm$

The length of the diagonal of a square is $8 cm$

Area of a square is given by $A = \frac {d^{2}}{2}$

Substituting $d = 8 cm$ 

$A = \frac {8^{2}}{2} = \frac {64}{2} = 32 cm^{2}$

Therefore, the area of a square of diagonal $8 cm$ is $32 cm^{2}$.

Ex 2: Find the length of the diagonal of a square of area $50 m^{2}$.

Area of square = $50 m^{2}$

$A = \frac {d^{2}}{2}$

Therefore, $50 = \frac {d^{2}}{2} =>\frac {d^{2}}{2} = 50 => d^{2} = 2 \times 50 => d^{2} = 100$

Taking the square root of both sides

$d = \sqrt{100} => d = 10 m$

Therefore, the length of the diagonal of a square whose area is $50 m^{2}$ is $10 m$.

Conclusion

The area of a 2D object is the region occupied by it in space. The area is measured in a square unit or $unit^{2}$. The area of a square depends on its side length and is calculated by finding the square of its side.

Practice Problems

  1. Find the area of a square whose side is
    1. $16 ft$
    2. $21 in$
  2. Find the length of the side of a square whose area is
    1. $225 mm^{2}$
    2. $529 m^{2}$
  3. Find the area of a square whose diagonal is
    1. $15 m$
    2. $56 m$
  4. Find the diagonal of a square whose area is
    1. $500 m^{2}$
    2. $512 m^{2}$
  5. How many square tiles of side $40 cm$ will be required to cover a square area of $100 m$?
  6. Find the cost of painting a square wall of side $15 m$ at the rate of ₹$5$ per $25 cm^{2}$.

Recommended Reading

FAQs

What is the formula for the area of a square?

The formula for area of a square of side $s$ is given by Area = $s^{2}$.

How do you calculate the area of a square?

The area of a square is calculated with the help of the formula: Area = $s^{2}$, where, $s$ is one side of the square. Since the area of a square is a two-dimensional quantity, it is always expressed in square units. For example, if we want to calculate the area of a square with a side $7$ unit, it will be Area = $7^{2} = 49 uni^{2}$.

How to find the area of a square from the diagonal of a square?

The area of a square can also be found if the diagonal of a square is given. The formula that is used in this case is: 

Area of a square = $\frac {d^{2}}{2}$, where $d$ is the diagonal of a square.

For example, the if the diagonal of a square is $9$ units, its area = $\frac {9^{2}}{2} = \frac {81}{2} = 40.5$ square unit.

What are the units of the area of a square?

Since the area of a square is a two-dimensional shape, it is always expressed in square units The common units of the area of a square are $m^{2}$, $in^{2}$, $cm^{2}$, $mm^{2}$.

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