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 in terms of the circumference: \( A(C) = \pi \left(\frac{C}{2\pi}\right)^2 \). Simplify this expression to complete the function.