To find the diameter of the circle with only the area known, use the formula \(A=\pi\times{r}^{2}\) where \(A=Area\), \(\pi≈3.1415926535897932\), \(r=radius\). We know that Area is equal to \({56.75in}^{2}\)and we know that \(\pi≈3.1415926535897932\), but we do not know what the radius is equal to. In order to figure out what the radius is equal to, solve for r and then plug in what we know.
\(A=\pi\times{r}^{2}\)
\(\frac{A}{\pi}=\frac{\pi\times{r}^{2}}{\pi}\)
\(\frac{A}{\pi}=1\times{r}^{2}\)
\(\frac{A}{\pi}={r}^{2}\)
\(\sqrt{\frac{A}{\pi}}=\sqrt{{r}^{2}}\)
\(\sqrt{\frac{A}{\pi}}=r\)
\(r=\sqrt{\frac{A}{\pi}}\)
\(r=\frac{\sqrt{A}}{\sqrt{\pi}}\)
\(r=\frac{\sqrt{A}}{\sqrt{\pi}}\times\frac{\sqrt{\pi}}{\sqrt{\pi}}\)
\(r=\frac{\sqrt{A}\sqrt{\pi}}{\sqrt{\pi}\sqrt{\pi}}\)
\(r=\frac{\sqrt{A\pi}}{\sqrt{\pi}\sqrt{\pi}}\)
\(r=\frac{\sqrt{A\pi}}{\sqrt{\pi\pi}}\)
\(r=\frac{\sqrt{A\pi}}{\sqrt{{\pi}^{2}}}\)
\(r=\frac{\sqrt{A\pi}}{\pi}\)
\(r=\frac{\sqrt{56.75\pi}}{\pi}\)
\(r≈\frac{\sqrt{56.75\times3.1415926535897932}}{3.1415926535897932}\)
\(r≈\frac{\sqrt{178.2853830912207641}}{3.1415926535897932}\)
\(r≈\frac{13.3523549642458488717}{3.1415926535897932}\)
\(r≈4.2501865889546685678486\)
Now we know that the radius is approximately equal to \(4.2501865889546685678486\); however, we want to know what the diamanter of the circle is, not the radius. In order to find the diamater, use the formula \(d=2r\) where \(d=diamater\), \(r=radius\).
\(d=2r\)
\(d≈2\times4.2501865889546685678486\)
\(d≈8.5003731779093371356972\)
The diameter of a circle wth an area of \({56.75in}^{2}\) is approximately equal to \(8.5003731779093371356972\).
