How Does the Diagonal Help Calculate the Area of a Square?
Area of any geometrical figure is the space occupied by a two-dimensional object. A square is a two-dimensional geometric shape that is determined by the sides which are all equal in length and perpendicular to each other (angle between two sides is 90 degrees). The area of a square is the number of square units required to fill a square fully. There are different ways to calculate the area of a square.
One of the conventional and standard ways to calculate the area of the square is by using its diagonals or by using its sides. Since all the sides of a square are the same, we can directly find the square of its side. Therefore, the area of a square is equal to the product of any of its two sides.
But sometimes the length of the side is not given and all we know is the length of the square's diagonal. With the knowledge of right triangles, we can find the area of a square using diagonal.
What is the Diagonal of a Square?
A diagonal is a line that stretches from one corner of a figure to the opposite corner, passing through the center of the figure. The diagonals of a square are always equal to each other. In a polygon, the diagonals can be defined as a line joining its two nonadjacent vertices.
The Relation Between Diagonal and Side of Square
A square can be divided into two right triangles, where the diagonal of the square is equal to the length of the hypotenuse of the triangle. Pythagoras theorem, which applies to right-angled triangles, shows the relation between the hypotenuse and sides of a right triangle.
Thus, it also represents the relation between the diagonal of a square (the hypotenuse of the triangle) and its sides.
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Using Pythagoras Theorem,
(Hypotenuse)2= (Base)2+ (Perpendicular)2
Here, the length of the base is equal to the length of perpendicular which is denoted by ‘a’ and hypotenuse is equal to the diagonal which is denoted by ‘d’.
Therefore, a2+ a2 = d2
Diagonal = \[\sqrt{a^{2} + a^{2}}\]
= \[\sqrt{2a^{2}}\]
= \[a\sqrt{2}\]
= side \[\sqrt{2}\]
Formula of the Area of Square Using Diagonal
Using the length of the diagonal, the area of a square can be calculated as:
Area of square = ½ × d2 units2
Here, “d” is the length of any of the diagonals. Also, remember that in a square, diagonals are equal.
Derivation of the Area of Square Using Diagonal
We know the formula to find the area of a square using diagonals. Now, we will derive that formula using the following two methods.
Using Pythagoras Theorem
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In the given figure, the diagonal of length “d” units divide the square of the side “a” units into two right triangles. Now, applying Pythagoras theorem in any right-angled triangle,
(Hypotenuse)2= (Base)2+ (Perpendicular)2
Here,
Perpendicular = a
Base = a
Hypotenuse = d
So,
a2+ a2 = d2
2a2 = d2
Or, a2 = d2/2
We know that area of a square = a2 = d2/2
Thus, area of a square using diagonals = ½ × d2 square units.
Using Relation between Side and Diagonal
For a square of side length ‘a’ and diagonal length ‘d’, we know,
Area of a square = side x side = a2
Now, as we have derived above,
Diagonal of square = side x √2 = a√2
Then, side of square, a = 1/√2 x diagonal = d/√2
Thus, area of square = a2
Area = (d/√2)2
Area = d2/2
Area = ½ x d2
Area = ½ x (diagonal)2
Thus, area of a square using diagonals = ½ × d2 square units.
Solved Examples
Example 1: Find the sides and area of a square when diagonal is given as 6 cm.
Solution: Let us take a square of side x. If the square is divided into two right-angled triangles then the hypotenuse of each triangle is equal to the diagonal of the square. As given, the diagonal is equal to 6 cm.
According to Pythagoras theorem,
x2 + x2 = 62
2x2 = 36
x2 = 18
x =\[\sqrt{18}\]
x = 3\[\sqrt{2}\] units
Thus, the length of the side of a square is 3\[\sqrt{2}\] units.
To find the area of a square, when diagonal is given, we can use any of the below methods:
Method 1
Area of a square = side x side =3\[\sqrt{2}\] x 3\[\sqrt{2}\]
= 9 x 2 = 18 cm2
Method 2
Area of a square = ½ x d2 = ½ x 6 x 6
= ½ x 36 = 18 cm2
Example 2: Find the length of the diagonal of a square using the Pythagoras theorem if the sides are 4 cm.
Solution: We know that all the sides of a square are equal in length. We also know that each vertex makes an angle of 90°. Now, let’s split the square into two right triangles, with sides equal to 4 cm. Using the Pythagoras theorem in one of the triangles, we will find the third side of the triangle, which is the diagonal of the square.
Let the hypotenuse/ diagonal be ‘c’ cm.
Therefore, (Hypotenuse)2= (Base)2+ (Perpendicular)2
c2= 42 + 42
= c2= 16 + 16
c = \[\sqrt{32}\] cm
c = 4\[\sqrt{2}\]cm
The length of the diagonal is 4\[\sqrt{2}\]cm
FAQs on Area of Square Using Diagonal: Formula & Step-by-Step Guide
1. What is the formula for calculating the area of a square using its diagonal?
The formula to find the area of a square when the length of its diagonal is known is: Area = (d²) / 2, where 'd' represents the length of the diagonal. This formula allows you to calculate the area without needing to know the side length.
2. How do you calculate the area of a square step-by-step if you only know its diagonal?
To find the area of a square using only its diagonal, follow these simple steps:
Step 1: Identify the length of the diagonal (d).
Step 2: Square this length, which means multiplying the diagonal's length by itself (d x d or d²).
Step 3: Divide the result from Step 2 by two.
Step 4: The final value is the area of the square, expressed in square units (e.g., cm², m²).
3. Why is the area of a square's formula (d²)/2? Can you explain its derivation?
This formula is derived using the Pythagorean theorem. A square's diagonal divides it into two identical right-angled isosceles triangles. The diagonal acts as the hypotenuse, and the two sides of the square ('a') are the other two sides of the triangle. According to the Pythagorean theorem: a² + a² = d². This simplifies to 2a² = d². Since the area of the square is a², we can rearrange the equation to a² = d²/2. Therefore, the area of a square can be directly calculated as half of the square of its diagonal.
4. How is the formula for the area of a square using its diagonal related to the area formula of a rhombus?
The relationship exists because a square is a special type of rhombus—one where all angles are 90° and the diagonals are equal in length. The area of any rhombus is calculated as (p × q) / 2, where 'p' and 'q' are the lengths of its two diagonals. In a square, both diagonals are equal (p = q = d). If we substitute 'd' for both 'p' and 'q' in the rhombus formula, we get (d × d) / 2, which simplifies to d²/2, the specific formula for the area of a square using its diagonal.
5. When is it more practical to use the diagonal formula for a square's area instead of the side-length formula (side²)?
It is more practical to use the diagonal formula (d²/2) in situations where:
The length of the diagonal is given directly in a problem, and the side length is unknown.
In real-world applications like construction or carpentry, it might be easier to measure the diagonal across a space than to measure a side precisely, especially if corners are obstructed.
The problem involves concepts from coordinate geometry where the diagonal's endpoints are known, making it easy to calculate its length using the distance formula.
6. If you double the length of a square's diagonal, does its area also double?
No, the area does not double. If you double the diagonal's length, the area becomes four times larger. This is because the area is proportional to the square of the diagonal (d²). Let the original area be A = d²/2. If the new diagonal is 2d, the new area will be (2d)²/2 = (4d²)/2 = 4 × (d²/2). Thus, the new area is four times the original area.
7. What is a diagonal of a square, and what are its most important properties?
A diagonal of a square is a line segment that connects two opposite vertices (corners). Every square has two diagonals with several key properties:
Equal Length: The two diagonals of a square are always equal in length.
Perpendicular Bisectors: They cut each other in half (bisect) at a perfect 90-degree angle at the centre of the square.
Angle Bisection: Each diagonal bisects the corner angles (90°) into two 45° angles.
