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Chapter 1: Problem 64

Write the area \(A\) of a circle as a function of its circumference \(C\).

Short Answer

Expert verified
The area \(A\) of a circle as a function of its circumference \(C\) is given by \(A = \frac{C^{2}}{4*\pi}\).

Step by step solution

01

Identify the Formulas Required

Firstly, you need to understand the formulas related to a circle. The formula to calculate the area of a circle is \(A = \pi * r^{2}\), where \(A\) represents the area, \(r\) represents the radius, and \(\pi\) is a constant approximately equal to 3.1416. Additionally, the formula for calculating the circumference of a circle is \(C = 2*\pi*r\), where \(C\) represents the circumference, \(r\) represents the radius, and \(\pi\) is a constant.
02

Isolate the radius

From the formula of the circle's circumference \(C = 2*\pi*r\), you wish to isolate \(r\), as this will help you to form the area as a function of the circumference. Here, you rearrange for \(r\) and find that \(r = \frac{C}{2*\pi}\).
03

Substitute the radius into the area formula

Using the isolated radius expression from Step 2, substitute \(r\) into the area formula \(A = \pi * r^{2}\). This gives \(A = \pi * (\frac{C}{2*\pi})^{2}\).
04

Simplify the expression

It is necessary to simplify the area expression from Step 3. The result will be \(A = \frac{C^{2}}{4*\pi}\). Hence, the problem of expressing the area of a circle as a function of its circumference is solved.

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