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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.

Step $$2$$: Draw a line segment $$AB$$ of length $$5$$ $$cm$$.

Step $$3$$: With $$A$$ as centre, draw a perpendicular line.

Step $$4$$: With $$A$$ as centre and with $$5$$ $$cm$$ as radius, draw an arc on the perpendicular line. Mark the intersection as $$D$$.

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 $$C$$.

Step $$6$$: Join $$BC$$ and $$CD$$ to form the desired square.

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.

Step $$2$$: Draw a line segment $$AC$$ of length $$10$$ $$cm$$.

Step $$3$$: Draw a perpendicular bisector to $$AC$$ such that the bisector intersects $$AC$$ at $$O$$.

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 $$D$$.

Step $$5$$: Join $$AB$$, $$BC$$, $$CD$$, and $$AD$$ to form the desired square.

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, $$\text{Area of a square} = 7.1^2$$

$$= 50.41$$ $$cm^2$$