Question

If the area of the circle is numerically equal to twice its circumference then what is the diameter of the circle? [CBSE 2011]

Answer 1

Let the diameter of the required circle be d.
Now, Area of circle = 2 ⨯ Circumference of the circle
$$\begin{gathered} \Rightarrow \mathrm{\pi }{\left(\frac{d}{2}\right)}^2=2\left(2\mathrm{\pi }\times \frac{d}{2}\right) \\ \Rightarrow {\left(\frac{d}{2}\right)}^2=2d \\ \Rightarrow d^2=8d \\ \Rightarrow d^2-8d=0 \\ \Rightarrow d\left(d-8\right)=0 \\ \Rightarrow d=8\mathrm{cm}\left[\because d\ne 0\right] \end{gathered}$$
Hence, the diameter of the of the circle is 8 cm.