What is the area of a circle with circumference of 9.42 inches?
+2441 To figure out this problem, we must know a few formulas that relate to a circle. Of course, you must know the area formula for a circle:
Let A= Area of a circle
Let r= radius of the circle:
\(A_{\circ}=\pi r^2\)
However, there is a problem currently, because we do not know the circumference of this circle! However, we can solve for the radius if we know the circumference. The formula for finding the circumference is:
Let C = circumference of a circle
Let r = radius
\(C_{\circ}=2\pi r\)
We know the circumference of the circle because it is given; the circumference is 9.42 inches. Let's solve for r:
| \(C_{\circ}=2\pi r\) | This is the formula for finding the circumference of a circle. We already know the circumference, so plug it into this equation |
| \(9.42=2\pi r\) | Divide by 2 on both sides of the equation |
| \(4.71=\pi r\) | Divide by \(\pi\) on both sides. |
| \(\frac{4.71}{\pi}=r\) | I am going to leave the r in this form because I want my final answer to be as exact as possible. |
Now that we know what r equals, we can substitute this into the area formula for a circle:
| \(A_{\circ}=\pi r^2\) | Substitute \(r\hspace{1mm}\text{for}\hspace{1mm}\frac{4.71}{\pi}\) |
| \(A_{\circ}=\pi (\frac{4.71}{\pi})^2\) | We could simply input this into the calculator, but we can actually simplify this further. Let's do it! First, do \((\frac{4.71}{\pi})^2\). Remember that the exponent is distributed to both the numerator and denominator |
| \(A_{\circ}=\frac{\pi}{1}*\frac{22.1841}{\pi^2}\) | Multiply the fractions together. |
| \(A_{\circ}=\frac{22.1841\pi}{\pi^2}\) | Now, we will utilize a fraction rule stating that \(\frac{a^b}{a^c}=a^{b-c}\) |
| \(A_{\circ}=22.1841\pi^{-1}\) | Now, we will use a power rule that says that \(a^b=\frac{1}{a^b}\hspace{3mm},b<0\) |
| \(A_{\circ}=\frac{22.1841}{\pi}\) | You cannot simplify further, so evaluate with a calculator now. |
| \(A_{\circ}=\frac{22.1841}{\pi}\approx7.0614in^2\) | Of course, keep units in your answer. |
