## Volume and Area of Solid In Terms of One Length For Lengths in Proportion

We can find an expression for the surface are of a simple solid shape like a cuboid, cone or cylinder in terms of one length wherever the length of the shape in question are in a fixed proportion.
Suppose the height of a cuboid is twice its width and the length is three times its width.
If we label the with as
$x$
then the height is
$2x$
and the length is
$3x$
.

The volume is
$V=x \times 2x \times 3x=6x^3$
.
The surface area is
$A=2(x \times 2x + x \times 3x+2x \times 3x)=22x^2$
.
Suppise the height of a cone is twice the radius. We can write
$h=2r$
. The volume of the cone is
$V=\frac{1}{3} \pi r^2 h= \frac{1}{3} \pi r^2 (2r)= \frac{2}{3} \pi r^3$
.
The surface area is
\begin{aligned} A &= \pi r(r+\sqrt{r^2+h^2}) \\ &=\pi r (r+\sqrt{r^2+(2r)^2} ) \\ &= \pi r (r+\sqrt{5r^2}) \\ &= \pi r^2 (1+ \sqrt{5}) \end{aligned}