If the radius of a cylinder is twice its height, write a formula for the surface area in terms of its height only.
My textbook says the answer is $12\pi h^2,$ but I got a completely different answer if someone could work me through this it would be a great help, thank you.
From wikipedia, $S = 2πrh + 2πr^2$
If $r = 2h$, we can substitute in: $S = 2πrh + 2πr^2 = 2π(2h)h + 2π(2h)^2 = 4πh^2 + 8πh^2 = 12πh^2$
suppose the height of the cylinder is $h.$ then the radius is $2h$ and the circumference of the cylinder is $2 \pi r = 4\pi h$ the curved surface area is $4 \pi h h = 4\pi h^2$
the lid and the bottom circles have the area twice $\pi (2h)^2 = 4\pi h^2$ all together you have $12 \pi h^2$ in surface area.
Let $r$ denote the radius and $h$ denote the height. If the radius of a cylinder is twice its height, then $r = 2h$.
Now, use the formula $$ A = 2 \pi r h + 2 \pi r^2 $$ substituting $2h$ for $r$.
Let us denote by $R$ the radius of the cylinder and $h$ its height : we thus have $2h=R$ . The area of one disc is $$S_{D}=\pi R^{2}$$ and the area of the lateral surface is $$S_{L}=2\pi Rh.$$ Thus the total area is $$2S_{D}+S_{l}=2\pi R^{2}+2\pi Rh=8\pi h^{2}+4\pi h^{2}=12\pi h^{2}$$ as claimed.
