### How to increase volume of a cylinder while keeping its surface area the same

Yesterday I noticed that someone searching the Web using the phrase “how do you make the volume of a cylinder greater without increasing the surface area” came to this blog. I didn’t check which specific post had gotten the hit, but it was probably this one. That post, however, didn’t answer the subject question. Then I got curious and decided to work out the answer.

Another way to state the question is can you increase volume of a cylinder while keeping its surface area the same? The answer is yes, one can increase the volume of a cylinder while keeping its surface area the same. One can do that only by increasing the cylinder's height (H) so that the volume (V) will increase, while at the same time, decreasing its radius (R) to keep the surface area (S) the same or by increasing R and decreasing H.

All we need are the following equations for V and S.

V=πR^{2}H

S=2πRH + 2πR^{2}

Let us look at the case when we increase H and decrease R. To get the new height (H_{n}) we multiply the old height (H_{o}) by a constant c>1 and to get the new radius (R_{n}) we multiply R_{o} by a constant k<1.

H_{n} = cH_{o}

R_{n} = kR_{o}

The new volume and the new surface area are

V_{n} = πR_{n}^{2}H_{n} = ckπR_{o}^{2}H_{o}

S_{n}=2πR_{n}H_{n} + 2πR_{n}^{2} = ck2πR_{o}H_{o} + k^{2}2πR_{o}^{2}

The required condition S_{n} / S_{o} = 1 now becomes after proper substitutions:

S_{n} / S_{o} = [(ck)(2πR_{o}H_{o}) + k^{2}(2πR_{o}^{2})] / [(2πR_{o}H_{o}) + 2πR_{o}^{2})] = 1

Let's simplify this by assuming that R_{o} = H_{o} = 1

S_{n} / S_{o} = (kc + k^{2}) / 2 = 1

By rearranging, we get a relationship between c and k:

c = (2 - k^{2}) / k

Remember that this is good only when R_{o} = H_{o} = 1.

Now let's try a numerical example. If we let c = 2, we get the following quadratic equation:

k^{2} + 2k - 2 = 0

The positive root is 0.732 (quadratic equation calculator). If we plug all the numbers in the appropriate equations and ignore the rounding off errors, we will get:

V_{o} = π

V_{n} = 1.464π

S_{o} = 4π

S_{n} = 4π

We have increased the volume while keeping the surface area the same.

Here is a new question: Is it possible to increase the volume of a cylinder while reducing its surface area?

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