Theory:

A square can be constructed using a few known measurements.
 
The measurements are:
1. When the side is known.
 
2. When the diagonal is known
Method \(1\): When the side is known
Let us construct a square with \(5\) \(cm\) as its side. Let us also find the area of the square obtained.
 
Step \(1\): Draw a rough diagram for the measurements given.
 
1324_42.svg
 
Step \(2\): Draw a line segment \(AB\) of length \(5\) \(cm\).
 
1324_43.svg
 
Step \(3\): With \(A\) as centre, draw a perpendicular line.
 
1324_44.svg
 
Step \(4\): With \(A\) as centre and with \(5\) \(cm\) as radius, draw an arc on the perpendicular line. Mark the intersection as \(D\).
 
1324_45.svg
 
Step \(5\): With \(D\) as centre and with \(5\) \(cm\) as radius, draw an arc. Similarly, with \(B\) as centre and with \(5\) \(cm\) as radius, cut the existing arc. Mark the intersection as \(C\).
 
1324_46.svg
 
Step \(6\): Join \(BC\) and \(CD\) to form the desired square.
 
1324_47.svg
 
To find the area of the square:
 
\(\text{Area of a square} = \text{Side} \times \text{Side}\)
 
\(\text{Area of a square} = \text{Side}^2\)
 
Here, \(\text{Side} = 5\) \(cm\).
 
Therefore, \(\text{Area of a square} = 5^2\)
 
\(= 25\) \(cm^2\)
Method \(2\): When the diagonal is known
Let us construct a square with one of its diagonals as \(10\) \(cm\). Let us also find the area of the square obtained.
 
Step \(1\): Draw a rough diagram with the measurements known.
 
1324_52.svg
 
Step \(2\): Draw a line segment \(AC\) of length \(10\) \(cm\).
 
1324_48.svg
 
Step \(3\): Draw a perpendicular bisector to \(AC\) such that the bisector intersects \(AC\) at \(O\).
 
1324_49.svg
 
Step \(4\): With \(O\) as centre and with \(5\) \(cm\) as radius, draw arcs on both sides of the perpendicular bisector. Mark the intersections as \(B\) and \(D\).
 
1324_50.svg
 
Step \(5\): Join \(AB\), \(BC\), \(CD\), and \(AD\) to form the desired square.
 
1324_51 (1).svg
 
To find the area of the square:
 
\(\text{Area of a square} = \text{Side} \times \text{Side}\)
 
\(\text{Area of a square} = \text{Side}^2\)
 
Here, side is unknown.
 
Therefore, we should measure the length of the side manually.
 
On measuring, we found that, \(\text{Side}\) \(=\) \(7.1\) \(cm\)
 
Now, \(\text{Area of a square} = 7.1^2\)
 
\(= 50.41\) \(cm^2\)