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  
  then the height is  
  and the length is  

cuboid with sides in proportion

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  
. 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
\[\begin{equation} \begin{aligned} A &= \pi r(r+\sqrt{r^2+h^2}) \\ &=\pi r (r+\sqrt{r^2+(2r)^2} ) \\ &= \pi r (r+\sqrt{5r^2}) \\ &= \pi r^2 (1+ \sqrt{5}) \end{aligned} \end{equation}\]

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