Question

the circumference of a circle is 23π in. what is the area, in square inches? express your answer in terms of π. answer a = in²

Explanation:

Step1: Find the radius from the circumference formula

The formula for the circumference of a circle is \(C = 2\pi r\). Given \(C=23\pi\), we set up the equation \(2\pi r=23\pi\). Dividing both sides by \(2\pi\), we get \(r=\frac{23\pi}{2\pi}=\frac{23}{2}\) inches.

Step2: Calculate the area using the radius

The formula for the area of a circle is \(A=\pi r^{2}\). Substitute \(r = \frac{23}{2}\) into the formula: \(A=\pi(\frac{23}{2})^{2}\). Using the exponent rule \((\frac{a}{b})^{2}=\frac{a^{2}}{b^{2}}\), we have \(A=\pi\times\frac{23^{2}}{2^{2}}=\frac{529}{4}\pi\) \(in^{2}\).

Answer:

\(\frac{529}{4}\pi\)