To find a function that models the area A of a circle in terms of its circumference C, we can start by recalling the formulas for the area and circumference of a circle:
- The area of a circle is given by the formula: A = πr²
- The circumference of a circle is given by the formula: C = 2πr
Where r represents the radius of the circle. To express the area in terms of the circumference, we first need to solve the circumference formula for r:
- Rearranging the circumference formula, we have:
- r = C / (2π)
Now that we have r in terms of C, we can substitute this expression into the area formula:
- Substituting for r, the area becomes:
- A = π(C / (2π))²
- Expanding this gives:
- A = π(C² / (4π²))
- Finally, simplifying this results in:
- A = C² / (4π)
Thus, the area of a circle can be modeled as a function of its circumference C by the formula:
A(C) = C² / (4π)
This formula indicates that if you know the circumference of a circle, you can easily calculate its area using this relationship. This is particularly useful in various applications in geometry and real-world scenarios where one measurement might be more readily available than the other.