Question

A right circular cylinder whose radius $r$is half of its height

Short answer

(a) Volume of the circular cylinder whose radius $r$is half of its height is,$V=\frac{{\mathrm{\pi h}}^3}{4}$.

(b) Surface area of the circular cylinder whose radius $r$is half of its height is,$S=2{\mathrm{\pi h}}^2$.

Step-by-step solution

 Part (a) step 1. Given information  

We have to find the volume and surface area of the given figure.

For that, we have been given a circular cylinder whose radius $r$is half of its height

Therefore,

$r=\frac{h}{2}$

Part (a) step 2. The volume of the circular cylinder 

The formula to find the volume of the circular cylinder of radius $r$and height $h$is,$V={\mathrm{\pi r}}^2\mathrm{h}$

By substituting the value of radius in this equation we get,

$$\begin{gathered} V=\pi \times r^2\times \mathrm{h} \\ =\mathrm{\pi }\times {\left(\frac{\mathrm{h}}{2}\right)}^2\times \mathrm{h} \\ =\frac{{\mathrm{\pi h}}^3}{4} \end{gathered}$$

Hence the volume of the circular cylinder whose radius $r$is half of its height is, $V=\frac{{\mathrm{\pi h}}^3}{4}$

Part (b) step 3. The surface area of the circular cylinder  

The formula to find the surface area of the circular cylinder is, $S=2\pi rh+2\pi r^2$

By substituting the value of radius in this equation we get,

$$\begin{gathered} S=2\times \mathrm{\pi }\times \frac{\mathrm{h}}{2}\times \mathrm{h}+2\times \mathrm{\pi }\times {\left(\frac{\mathrm{h}}{2}\right)}^2 \\ =\mathrm{\pi }\times {\mathrm{h}}^2+\mathrm{\pi }\times {\mathrm{h}}^2 \\ =2{\mathrm{\pi h}}^2 \end{gathered}$$

Hence the surface area of the circular cylinder whose radius $r$is half of its height is,$S=2{\mathrm{\pi h}}^2$