The total distance around the outside of a square of the length of the boundary of a square is known as the Perimeter of a square. The Perimeter of a polygon is one of the easiest to find as one would obtain the Perimeter if they were to add the length of each of the sides. 

Hence for a rectangle, the perimeter would be equal to length + breadth + length + breadth, that is, (2 x length) + (2 x breadth). The square is a rectangle with four equal sides, and this equation can be remodified to obtain the Perimeter of a square in the following way: –

The Perimeter of a square = Length of one side + Length of one side + Length of one side + Length of one side 

Which can further be simplified to: –

The Perimeter of a square = 4 x Length of one side (S), that is, Perimeter of a square = 4S (Where S is the length of any one of the sides)

However, this method can only be used if all the lengths of all the sides are known. 

The following method explains how the Perimeter of a square can be found in the area is known.

1. The Area of a rectangle can be found by the formula length x breadth. Here as the length and breadth are the same on the case of a square, the Area of a square would be equal to side x side, that is Area of a square = S2

The square root of the Area would give the length of the side of the square which can further be used to find the perimeter of the square by using the above-given equation (Perimeter of a square = 4S)

The Perimeter of a square inscribed inside a circle can be found out in the following way: –

1. When a square is inscribed inside a circle, all four corners of the square lie on the edge of the circle. The distance from the center of the inscribed circle to one of the corners that touch the edge of the circle would be the radius of the circle. Twice, the radius of a circle would give the diameter of a circle. This diameter can be used to divide the square into two triangles with the diameter acting as the hypotenuse of both the triangles.

Therefore, the triangles would have alongside a, shorter side b, and a hypotenuse c wherein c = 2r (2 x radius). Using the Pythagoras theorem, a2 + b2 = c2, here as the triangles are sides of a square, both a and b would be equal (a=b) and hence 2 x a2 = c2 where 2a2 = (2r)2. Therefore, 2a2 = 4r2. 

After simplification, a2 = 2r2. And from this, a can be found from the square root of 2r2, (a = √2r2). And multiplying this a with four would give the Perimeter of the square (Perimeter of a square = 4a)

These are the three different methods through which the Perimeter of a square can be found.