Functions and Graphs


Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.

Verified step by step guidance

1. Start by recalling the formulas for the area and circumference of a circle. The area \( A \) of a circle is given by \( A = \pi r^2 \), where \( r \) is the radius. The circumference \( C \) is given by \( C = 2\pi r \).
2. Express the area \( A \) as a function of the radius \( r \). This is already given by the formula \( A(r) = \pi r^2 \).
3. Express the circumference \( C \) as a function of the radius \( r \). This is given by the formula \( C(r) = 2\pi r \).
4. To express the area as a function of the circumference, first solve the circumference formula for \( r \). From \( C = 2\pi r \), we get \( r = \frac{C}{2\pi} \).
5. Substitute \( r = \frac{C}{2\pi} \) into the area formula \( A = \pi r^2 \) to express the area as a function of the circumference: \( A(C) = \pi \left(\frac{C}{2\pi}\right)^2 \). Simplify this expression to complete the function.

Key Concepts

Area of a Circle

The area of a circle is calculated using the formula A = πr², where A represents the area and r is the radius. This formula shows that the area is directly proportional to the square of the radius, meaning that as the radius increases, the area increases quadratically.

Circumference of a Circle

The circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. This formula indicates that the circumference is directly proportional to the radius, meaning that if the radius doubles, the circumference also doubles.

Function Relationships

In this context, expressing the area as a function of the circumference involves manipulating the formulas for area and circumference. By substituting the expression for radius from the circumference formula into the area formula, we can derive a new function A(C) that relates area directly to circumference.