What is a Square?
A square is a quadrilateral that has four sides that are equal in length. Additionally, all the four interior angles in a square are \( 90^{\circ} \). The diagonals of a square are equal and they also bisect each other at right angles.
Formula for the Diagonal of a Square
A line joining the opposite vertices of a square is called the diagonal of the square.
A square has 2 diagonals of the same length. If the side length of a square is ‘\( a \)’ units then the diagonal ‘\( d \)’ of the square is represented by the formula,
\( d=\sqrt{2}a \) units
So, the diagonal of a square is \( \sqrt{2} \) times its side length, and the length of the diagonal is measured in the units of length.
Formula for the Perimeter of a Square
The perimeter of a square is the total length of its boundary. The length of the boundary of a square is measured by adding up the lengths of its four sides.
If the side length of a square is ‘\( a \)’ units then its perimeter will be given by the formula:
Perimeter of a square, \( P = a + a + a + a = 4a \)
Here, \( P \) denotes the perimeter and \( a \) denotes the side length of the square.
So, the perimeter of a square is four times its side length.
Perimeter is measured in the units of length.
Formula for the Area of a Square
The area of a square is the measure of the region enclosed by its boundary.
If the side length of a square is ‘\( a \)’ units then its area will be given by the formula:
Area of a square, \( A = a \times a = a^2 \)
Here, \( A \) denotes the area and \( a \) denotes the side length of the square.
So, the area of a square is the square of its side length.
Area is measured in square units.
Solved Examples
Example 1: Find the length of the diagonal of the given square.
Solution:
Side of the square, \( a=14 \) cm
\( d=\sqrt{2}a \) Formula for the diagonal of a square
\( ~~=\sqrt{2}\times 14 \) Substitute \( 14 \) for \( a \)
\( ~~=19.80 \) Multiply
Therefore the diagonal of given square measures \( 19.80 \) cm.
Example 2: The diagonal of square is \( 11.31 \)cm. Find the area of the square.
Solution:
The diagonal of square is \( 11.31 \) cm.
To calculate the area, we need the side length of the square.
Let us apply the diagonal formula to calculate the side length:
\( d=\sqrt{2}a \) Formula for the diagonal of a square
\( 11.31=\sqrt{2}\times a \) Substitute \( 11.31 \) for \( d \)
\( a=\frac{11.31}{\sqrt{2}} \) Solve for \( a \)
\( a=8 \) Simplify
So the side of square is \( 8 \) cm.
\( A=a^2 \) Formula for the area of a square
\( ~~~=8\times 8 \) Substitute \( 8 \) for \( a \)
\( ~~~=64 \)
Therefore the area of the square is \( 64 \) square centimeters.
Example 3: Peter owns a square field. The side length of the field is \( 850 \) meters. Find the cost of fencing the field at a rate of \( \$ 0.05 \) per meter of fencing wire.
Solution:
To determine the total cost of fencing, we first need to calculate the total length of boundary of the crop field, that is, its perimeter.
The field is square in shape so:
\( P=4a \) Formula for the perimeter of a square
\( ~~~=4\times 850 \) Substitute \( 850 \) for \( a \)
\( ~~~=3400 \)
The total cost of fencing the boundary will be the total boundary length multiplied by the rate of fencing, that is,
Total cost of fencing \( = \) Total boundary length \( \times \) Rate of fencing
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=3400\times 0.05 \)
\( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=170 \)
Therefore the total cost of fencing the crop field is \( \$ 170 \).
A square is a two dimensional closed figure with four sides. In a square all four sides are equal in length and all four interior angles are right angles.
The line that joins the opposite vertices of a square is called the diagonal of the square.
In a square all the sides are equal in length, but in a rectangle only the opposite sides have equal lengths.