Square
In geometry, a square is a regular quadrilateral that has four equal sides and four angles each of which are 90\(^{\circ}\) in measure. As a result, the sum total of the internal angles in a square is 360\(^{\circ}\). A square has two diagonals, and as seen earlier, are created by connecting the opposing vertices of a square.
Properties of Diagonals
In order to list the qualities of diagonals, consider the following square shown in the image.
- The diagonals of a square are of equal length
- Diagonals of a square bisect each other perpendicularly
- They divide the square into two identical, right-angled triangles using isosceles triangles
- The mid point of both diagonals is where they intersect, which means that both diagonals bisect each other into two halves
How is the Formula for the Diagonal of a Square derived?
Let’s take the triangle ADC. Since, we are aware that every angle in a square is 90° degrees, we can apply the Pythagoras theorem to determine the hypotenuse, which in this instance is ‘\(d\)’.
\(d^2=a^2\ +\ a^2\)
\(d=\sqrt{a^2+a^2}\) (Bringing the square from L.H.S to R.H.S)
\(d=\sqrt{{2a}^2}\) (Simplified)
\(d=a\sqrt2\)
Formula for the Diagonals of a Square
Diagonal of a square formula is, \(d=a\sqrt2\)
Where ‘\(a\)’ is a side of the square and ‘\(d\)’ is the diagonal.
The Pythagorean theorem is used to calculate the formula for the diagonal of a square.
As seen earlier on in this article, the diagonal creates two isosceles right-angled triangles in a square. Both diagonals cut each other at right angles and are congruent.
Solved Examples
Example 1: If a square has an area of 25 square units, determine the length of its diagonal?
Solution:
Given, area of the square = 25 square units
\(a^2=25\) [Use the formula for the area of a square]
Therefore, \(a=\sqrt{25}=5\) units
Since, \(d=a\sqrt2\) [Use the formula of diagonals of a square]
\(=a\sqrt2=5\sqrt2=7.071\) units
Hence, the length of the diagonal of a square is 7.071 units.
Example 2: If a square has an area of 64 square units, determine the length of the diagonal.
Solution:
Given: Area of the square = 64 square units
\(a^2=64\) [Use the formula for the area of a square]
Therefore, \(a=\sqrt{64}=8\) units
Since, \(d=a\sqrt2\) [Use the formula of diagonals of the square]
\(=a\sqrt2=8\sqrt2=11.312\) units
Hence, the length of the diagonal of the square is 11.312 units.
Example 3: Rita has a square sheet of paper with a side length of 4 inches. She wants to decorate the diagonals of this sheet with colorful sticker paper. What is the total length of the sticker paper she will need for both the diagonals?
Solution:
Given: Side of the square = 4 inches
Since, \(d=a\sqrt2\) [Use the formula of diagonals of the square]
\(=a\sqrt2=4\sqrt2=5.656\) inches
Hence, the length of the sticker paper needed is twice the length of one diagonal, \(5.656\times 2 = 11.31\) or approximately 11.5 inches.
Every square has two diagonals that are of equal length. These diagonals 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 the diagonal. Hence the side length cannot be longer than the diagonals of a square.
The diagonal is root two times the side length of a square.