Question

Prove, A = $\frac{1}{2}$ C r, for a circle having radius, circumference and area r, C, A respectively.

Answer 1

We have to prove that A = $\frac{1}{2}\mathrm{C}r$, where A, C and r are the area, circumference and radius of the circle, respectively.
We know:
Area of a circle = $\mathrm{\pi }{r}^2$
$$\begin{gathered} \Rightarrow A=\mathrm{\pi }{r}^{\mathit{2}}\mathit{}\mathit{} \\ \mathrm{On}\mathrm{multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}2,\mathrm{we}\mathrm{get}: \\ \Rightarrow 2\mathrm{A}=2\mathrm{\pi }{r}^{\mathit{2}} \\ \Rightarrow 2\mathrm{A}=\left(2\mathrm{\pi }r\right)\times r \\ \Rightarrow 2\mathrm{A}=\mathrm{C}\times \mathrm{r}[\mathrm{C}=2\mathrm{\pi }r] \\ \Rightarrow \mathrm{A}=\frac{1}{2}\mathrm{Cr} \end{gathered}$$