# How to find the radius from the surface area of a cylinder?

• Jun 6th 2009, 03:01 PM
so774
How to find the radius from the surface area of a cylinder?
Hi. In my math class im learning about volume and surface area of a cylinder. There was one question on my homework that got me stuck.

How to find the height or radius from the given surface area?

Is there any formulas for finding it out?

Appericate the help thx!
• Jun 6th 2009, 03:22 PM
e^(i*pi)
Quote:

Originally Posted by so774
Hi. In my math class im learning about volume and surface area of a cylinder. There was one question on my homework that got me stuck.

How to find the height or radius from the given surface area?

Is there any formulas for finding it out?

Appericate the help thx!

A cylinder is two circles at the ends with a rectangular section folded around the centre.

Therefore surface area is given by $\displaystyle A = 4\pi r + 2\pi rh = 2\pi r(2+h)$ where r is the radius and h is the height of the cylinder. That can then be rearranged depending on whether you want to make r the subject or h the subject:

$\displaystyle r = \frac{A}{2\pi (2+h)}$

$\displaystyle h = \frac{A}{2\pi r} - 2$
• Jun 6th 2009, 09:51 PM
Prove It
Quote:

Originally Posted by e^(i*pi)
A cylinder is two circles at the ends with a rectangular section folded around the centre.

Therefore surface area is given by $\displaystyle A = 4\pi r + 2\pi rh = 2\pi r(2+h)$ where r is the radius and h is the height of the cylinder. That can then be rearranged depending on whether you want to make r the subject or h the subject:

$\displaystyle r = \frac{A}{2\pi (2+h)}$

$\displaystyle h = \frac{A}{2\pi r} - 2$

Wrong.

The net of a cylinder is two congruent circles and a rectangle.

The circles have radius $\displaystyle r$, so each has $\displaystyle A = \pi r^2$.

The rectangle has a width $\displaystyle h$ (the height of the cylinder) and the length is the circumference of the circle. So the length is $\displaystyle 2\pi r$. So the area of this rectangle is $\displaystyle 2\pi r h$.

Thus the Total Surface Area is given by

$\displaystyle TSA = 2\pi r^2 + 2\pi r h = 2\pi r(r + h)$.

Now you could rearrange the formula to give the radius or the height. The height is easy, radius is not so easy...

If $\displaystyle TSA = 2\pi r(r + h)$ then

$\displaystyle \frac{TSA}{2\pi r} = r + h$

$\displaystyle h = \frac{TSA}{2\pi r} - r$.

$\displaystyle TSA = 2\pi r^2 + 2\pi r h$

$\displaystyle 0 = 2\pi r^2 + 2\pi rh - TSA$

This is a Quadratic in $\displaystyle r$, so it is solved using the Quadratic Formula.

$\displaystyle r = \frac{-2\pi h \pm \sqrt{4 \pi^2 h^2 + 8\pi TSA}}{4 \pi}$

$\displaystyle = \frac{-2 \pi h \pm 2\sqrt{\pi^2 h^2 + 2\pi TSA}}{4\pi}$

$\displaystyle = \frac{-\pi h \pm \sqrt{\pi^2 h^2 + 2\pi TSA}}{2\pi}$

And since it's impossible to have a negative radius, the only solution that makes sense is

$\displaystyle r = \frac{\sqrt{\pi^2 h^2 + 2\pi TSA} - \pi h}{2\pi}$.