The circumference of a circle is measured as 28 cm with an error of 0.01 cm. The percentage error in the area, is

Let $r$ be the radius, $C$ the circumference and $A$ the

area of the circle. Then. $C=2\mathrm{\pi r}$ and $A={\mathrm{\pi r}}^2$.

$\begin{array}{l}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}A=\pi {\left(\frac{C}{2\pi }\right)}^2\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}A=\frac{c^2}{4\pi }\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\frac{dA}{dC}=\frac{C}{2\pi }\end{array}$

$\begin{array}{l}\therefore \mathrm{\Delta }A=\frac{dA}{dC}\mathrm{\Delta }C\\ \Rightarrow \mathrm{\Delta }A=\frac{C}{2\pi }\mathrm{\Delta }C\\ \Rightarrow \frac{\mathrm{\Delta }A}{A}\times 100=\frac{{\displaystyle \frac{c}{2\pi }}\times ∆c}{\frac{c^2}{4\pi }}\times 100=2\left(\frac{\mathrm{\Delta }C}{C}\times 100\right)\\ =2\times \frac{0.01}{28}\times 100=\frac{1}{14}\end{array}$