Understanding the Relationship Between Area and Perimeter of a Square
A square is a special type of quadrilateral where all four sides are equal in length. When we talk about the area and perimeter of a square, we can derive a function that links these two important measurements.
Formulas to Remember
- Side length (s): The length of one side of the square.
- Area (A): The area of a square is calculated using the formula:
A = s². - Perimeter (P): The perimeter of a square is calculated using the formula:
P = 4s.
Deriving Area as a Function of Perimeter
To express the area of the square as a function of its perimeter, start with the perimeter formula:
P = 4s
From this, we can solve for the side length s:
s = P / 4
Now, we can substitute this expression for s back into the area formula:
A = s² becomes A = (P / 4)².
When we simplify this, we get:
A = P² / 16
Final Result
Thus, the area A of a square can be expressed as a function of its perimeter P as follows:
A(P) = P² / 16
This relationship shows how the area increases quadratically with respect to the perimeter of the square.