Area of Square Using Diagonals: Formula to Calculate Area of a Square Using Diagonal

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The area of a square can be calculated using its diagonals. Other traditional method of finding the area of a square is square of the side, this is one of the most useful methods to find out the area of a square if the diagonal length is given.

Formula to Calculate Area of a Square Using Diagonal

For finding of area of the square is not always required to side. If we have the length of the diagonal then the area can be calculated as:

Here, “d” is the length of the diagonal of square.

Derivation for Area of Square Using Diagonal Formula

The formula to find the area of any square if its diagonals are given can be derived using Pythagoras theorem as explained below:

Consider a square of sides “a” units and diagonal as “d” units.

Derivation for Area of Square Using Diagonal

Derivation for Area of Square Using Diagonal

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In the above figure, the square of the side “a” unit, has been divided into two right triangles with the help of diagonal of length “d” units. Thus, the diagonal of the square divides it into two right triangles. Consider any right triangle and apply Pythagoras theorem.

According to Pythagoras theorem, for a right-angled triangle,

In the above diagram,

  • Perpendicular

  • Base

  • Hypotenuse

So,

Now, area of a square

So, area of a square using diagonals

Example Question to Solve Area of Square When Diagonal is Given:

Question: Find the area of a square having a diagonal of length

Solution:

Given the diagonal length,

Derivation for Area of Square Using Diagonal

Derivation for Area of Square Using Diagonal

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  • We have formula for finding the area of square when diagonal is given,

  • Now, the area of square

Hence the area of square is .

Note: Using the diagonal, the perimeter of the square can also be found as explained below.

Finding Perimeter of Square Using Diagonal

It is known that the area of a square in terms of diagonal

Now,

Or,

As the perimeter of square

Question: Find the perimeter of a square having a diagonal of length

Here given the diagonal length,

Finding Perimeter of Square Using Diagonal

Finding Perimeter of Square Using Diagonal

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We above discuss, how we calculate the perimeter of the square using diagonal and we find the formula,

  • Diagonal of the square

  • We put the value of diagonal in the formula,

100 is divided by 2.

Hence, the perimeter of the square is .