Area of a Square

The word “area” comes from the Latin area, which simply means vacant space. In the context of a square, the area of a square is simply the measure of space within square boundaries. In this article, we will not only be exploring every aspect of the area of a square but also its variations. 

1.0Introduction to Square

Let's learn the basics of this figure before knowing the concepts of the area of a square. A square is a quadrilateral with two dimensions that has all the same sides and angles, i.e., 90°. Because of the simplicity and symmetry of this shape, it is one of the first shapes we learn about in geometry. The shape also has two diagonals of the same measure, bisecting the square into two triangles of the same area, too.

2.0What is the Area of a Square? 

The square's area is the amount of total space occupied inside its four equal sides, which can be represented in square units, such as cm², m², or km². The general area of a square formula can be expressed as:  

$A=a\times a=a^2$

Here, 

• ‘A’ is denoted for the area of a square. 
• ‘a’ is denoted for the length of side of the given square. 

3.0Area of a Square Using Diagonal: 

In some instances, when the diagonal of a square is given instead of the side, how do you calculate the area of a square in this case? By measuring the side in terms of its diagonal. 

For this, simply use Pythagoras' theorem, as the diagonal divides the square into two right triangles, such that: 

AC = d = diagonal 

AD = DC = a = equal sides 

Using the theorem: 

d² = a² + a² = 2a² 

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

Now, substitute the value of a² in the area of a square formula, we will get: 

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

4.0Area of a Square Using Perimeter: 

Just like in the above case, if the perimeter of a square is given instead of its side. We can find the area of a square by finding the length of the side in terms of its perimeter. 

We know the perimeter of a square(P) = 4a 

$\frac{P}{4}$

Substituting the value of a in the area of a square formula: 

A = $(\frac{P}{4}{)}^2=\frac{P^2}{16}$

5.0How to Find the Area of a Square? 

How to find the Area of a Square solely depends on what information is provided in the question. That is: 

• If the side is given: Use the standard formula for the area of a square and multiply the side by itself.
• If the diagonal is given: Find the measure of the sides in terms of its diagonal using the specialised formula. 
• If the perimeter is given: Convert the perimeter of the square in terms of its side and then square it.

6.0Solved Examples

Problem 1: The cost of fencing a square garden at ₹120 per metre is ₹38,400. Find the area of a square garden. 

Solution: Given that: 

Per meter cost of fencing the garden: ₹120

Total cost of fencing the garden: ₹38,400

We know, the total cost of fencing = Perimeter Rate

38,400 = P x 120

$\frac{38400}{120}=320m$

Perimeter of a square garden = 4(side)

320 = 4(a)

a = 80m 

Now, the area of the square garden = a² 

A = (80)² = 6400m²

Problem 2: A square hall has a diagonal of 20√2 m. Square carpets of 2 m are used to cover the floor completely. Find how many carpets are needed.

Solution: Given that: 

Diagonal of the square hall = 20√2 m

Length of the side of the square carpet = 2m 

Now, 

Area of the hall $=\frac{d^2}{2}=\frac{(20\sqrt{2}{)}^2}{2}=\frac{800}{2}=400{\text{m}}^2$

Area of one square carpet = a² = (2)² = 4m² 

Number of carpets = Area of the square hall Area of one square carpet= $\frac{\text{Area of the square hall}}{\text{Area of one square carpet}}=\frac{400}{4}=100$

Therefore, 100 carpets are needed to cover the whole hall. 

Problem 3: A square photo frame has an outer side of 50 cm and an inner square cut out with a side of 30 cm. Find the area of the wood used in making the frame. 

Solution: Given that: 

Length of the outer side of the frame = 50cm 

Length of the inner side of the frame = 30cm 

Now, 

Area of the frame, including outer side = 50² = 2500cm²

Area of the frame, excluding outer side = 30² = 900cm²

Area of the wood used = 2500 – 900 = 1600cm² 

Problem 4: A square park has a diagonal of 70√2 m. A pathway of width 2 m is built inside the park along its boundary. Find the area of the pathway. 

Solution: Given that: 

The diagonal of the square park is 70√2m

The width of the pathway is 2m 

Therefore, the side of the square park in terms of its diagonal (a) $=\frac{d}{\sqrt{2}}=\frac{70\sqrt{2}}{\sqrt{2}}=70\mathrm{m}$

Hence, the side of the square park without the pathway = 70-4=66m

Now, 

The total area of the square park $=\frac{d^2}{2}=\frac{(70\sqrt{2}{)}^2}{2}=\frac{4900\times 2}{2}=4900{\mathrm{m}}^2$

The area of the square park without the pathway = a² = 66² = 4356m² 

Finally, the area of the path inside the park = 4900 – 4356 = 544m²