Solving for the Side of a Square When Its Perimeter Equals Its Area

Let the side length of a square be denoted as ( s ) centimeters. The perimeter ( P ) of a square is given by the formula:

( P 4s )

Meanwhile, the area ( A ) of a square is:

( A s^2 )

According to the problem, the perimeter of the square is equal to its area:

( 4s s^2 )

Solving the Equation

To find ( s ), we can rearrange the equation as follows:

( s^2 - 4s 0 )

This can be factored as:

( s(s - 4) 0 )

Solving for ( s ) gives us two solutions:

( s 0 ) and ( s 4 )

Since a side length cannot be zero, the side length of the square must be:

( boxed{4 , text{cm}} )

Verification

Let's verify the solution with the following conditions:

Side length of the square: 4 cm Perimeter: ( 4 times 4 16 , text{cm} ) Area: ( 4^2 16 , text{cm}^2 )

Both the perimeter and the area are equal, confirming that the side length is indeed 4 cm.

Alternative Approach

Another approach to solve the problem involves denoting the side of the square as ( x ) cm:

Perimeter: ( 4x ) cm Area: ( x^2 ) cm2

According to the problem:

( x^2 4x )

Solving for ( x ) gives:

( x^2 - 4x 0 )

This further simplifies to:

( x(x - 4) 0 )

Thus, the solutions are:

( x 0 ) and ( x 4 )

Again, since the side length cannot be zero, the side length of the square is:

( boxed{4 , text{cm}} )

Conclusion

In summary, if the perimeter of a square is equal to its area, the side length of the square must be 4 cm. This solution is consistent across various approaches and verifications.