What is the Diagonal of a Square?
The diagonals of a square are formed by connecting its opposite vertices. Every square has two diagonals. The properties of the diagonals are:
- Diagonals of squares are congruent
- They are perpendicular bisectors of each other
- Two congruent isosceles right-angled triangles are formed by a diagonal of the square
What is the Formula for the Diagonal of a Square?
The formula for the diagonal of a square is \( d=a\sqrt{2} \); where ‘d’ is the diagonal length and ‘a’ is the side length of the square. This formula is derived with the use of the Pythagorean theorem. Let’s see the manner in which the formula for the length of the diagonal is derived.
Derivation of the Diagonal of a Square Formula
Let’s consider the triangle ADC in the image and we can also see that the side length of the square ABCD is ‘a’. We already know that all the angles in a square are 90°.
Hence, by using the Pythagorean theorem, we can find the hypotenuse, which is ‘d’.
\( H^2=P^2+B^2 \) [Write Pythagorean theorem]
\( d^2=a^2+a^2 \) [Substitute d for H and a for both P and B]
\( d^2=2a^2 \) [Add]
\( \sqrt{d^2}=\sqrt{2a^2} \) [Apply square root to each side]
\( d=a\sqrt{2} \) [Simplify]
Therefore, the formula for the diagonal of a square is \( d=a\sqrt{2} \).
Area of Square by Applying Formula for the Diagonal
We can use the relationship between diagonal length d and the side a length of a square to find its area A.
\( A=a^2 \)
\( A=(\frac{d}{\sqrt{2}})^2 \) [From \( d=a\sqrt{2} \)]
\( A=\frac{d^2}{2} \)
So, the formula for area of square using diagonal is \( A=\frac{d^2}{2} \).
Rapid Recall
Area of a Square using Diagonal
Solved Examples
Example 1: If the side of a square is 15 inches, what would be the length of its diagonal?
Solution:
The side of a square 15 inches.
As we know, the formula to find the diagonal length of a square is,
\( d=a\sqrt{2} \) [Write the formula for diagonal]
\( d=15\sqrt{2} \) [Substitute 15 for a and 1.414 for \( \sqrt{2} \)]
\( d=21.21 \) [Multiply]
Thus, the length of the diagonal is 21.21 inches.
Example 2: On Paul’s birthday, his friends brought him a large square shaped cake as a birthday gift. If the diagonal length of the cake is 12 inches, can you find the actual size in terms of the area of the cake?
Solution:
In order to find the actual size of a cake, we need to find its area.
Thus, the area of a square can be calculated by the formula,
\( A=\frac{1}{2}\times d^2 \)
\( A=\frac{1}{2}\times 12^2 \) [Substitute the value of ‘d’ as 12]
\( A=\frac{1}{2}\times 144 \) [Find the square of 12]
\( A=\frac{144}{2} \) [Divide]
\( A=72 \)
Therefore, the actual size of Paul’s birthday cake is 72 square inches.
Example 3: Find the area of a square shaped parking lot whose diagonal is 20 feet in length.
Solution:
\( A=\frac{1}{2}\times d^2 \) [Write the formula for area of square]
\( A=\frac{1}{2}\times 20^2 \) [Substitute 20 for d]
\( A=\frac{1}{2}\times 400 \) [Find the square of 400]
\( A=\frac{400}{2} \) [Divide the values]
\( A=200 \)
So, the area of the plot is 200 square feet.
Every square has two diagonals which are of equal length and they also bisect each other at right angles.
The diagonals are the longest lines in a square.
In every right triangle we observe that the hypotenuse is the longest side of a triangle, and the diagonal is the hypotenuse of a triangle formed by two sides and a diagonal of square. Hence the diagonal cannot be longer than one of the sides of a square.
The diagonal is root two times the side length of a square.