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Calculating the area of a square with diagonals

The area of a square is the number of unit squares that can be fit into the square completely. To calculate the area of the square, the length and the breadth of the square are multiplied and the resulting product is the area of the square. However, we know that in a square the length and the breadth are the same, and hence the area of the square is a² sq units, where a = length of any of the sides of the square. However, sometimes the length of any side of the square might not be known and only the length of the diagonal of the square is made available. The length of the diagonals of a square is equal.

In this article, we will learn the formula to calculate the area of the square when the length of its diagonals is given, understand how that formula is derived, and go through some solved illustrations and examples.  

Formula for calculating the area of a square with diagonals:

The diagonal of the square is the line that connects any two opposite vertices of the square. The area of a square when only the diagonals are available =  (1/2) x d² sq units

Where d is the length of the diagonal.

Derivation

To understand the derivation of the formula, we must first remember the following

1. All the sides of the square are equal and are perpendicular to their adjacent sides
2. According to the Pythagoras theorem, (hypotenuse)² = (base)² + (perpendicular side)²

Now, in the above square ABCD, let us say the length of the diagonal BD is d and the length of the sides is ‘a’.  We can see that DC is perpendicular to BC and that ∆ BCD is a right-angle triangle.

From Pythagoras theorem, we know that (hypotenuse)² = (base)² + (perpendicular side)²

So, BD² = BC² + DC²

i.e d² = a² + a²

d² = 2a²

(1/2) x d² = a²

Solved Examples

1. Find the area of the square ABCD where the length of diagonal BD = 37 cm.

Solution:

We know that the area of the square where only the length of the diagonal is given is equal to  (1/2) x d² where d is the length of the diagonal.

So, the area of square ABCD  =  (1/2)  x 37² cm²

                                                                                                = 684.5 cm²

2. Find the side and the area of the square PQRS when the diagonal is 17 cm.

Solution:

We know from Pythagorean theorem, (hypotenuse)² = (base)² + (perpendicular side)²

If the side of the square is a cm, then

17² = a² + a2

2a² = 17²

a² =   (1/2) x 289

a = √144.5

a = side = 12.02 cm

Now that we know that the side = 12.02 cm, area of the square PQRS = 12.02 X 12.02 = 144.5 cm²

3. Find the diagonal if the sides of the square are 7 cm.

We know from Pythagoras theorem, (hypotenuse)² = (base)² + (perpendicular side)²

We know that in the given square, base = perpendicular side = 7 cm.

Hence, by substituting the above figures in the pythagoras theorem,

d² = 7² + 7²

d² =2x 7²

d =  √98

d = 9.89 cm