How to Compare the Volume of Two Cylinders by Scale Factors

Key takeaway: For cylinders, volume depends on both radius and height. That means one scale factor on the radius gets squared, while the height changes linearly. This is the part students miss most often.

The volume of a right circular cylinder is

$V=\pi r^2h$

If the second cylinder has volume 392 times the first, then the ratio must satisfy

$\frac{R^2H}{r^2h}=392$

Now test the answer choices:

• A: $R=8r$, $H=7h$

  $8^2\cdot 7=64\cdot 7=448$

• B: $R=8r$, $H=49h$

  $8^2\cdot 49=64\cdot 49$

• C: $R=7r$, $H=8h$

  $7^2\cdot 8=49\cdot 8=392$

• D: $R=49r$, $H=8h$

  $49^2\cdot 8$

Only choice C gives the correct ratio.

Correct answer: C

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Pitfalls the pros know 👇 Do not treat volume like area. For cylinders, the radius is squared, so doubling the radius multiplies volume by 4, not 2. Another easy mistake is checking only one dimension and forgetting the other one entirely.

What if the problem changes? If the volume ratio were $k$, you would need values of $R$ and $H$ such that

$\left(\frac{R}{r}\right)^2\left(\frac{H}{h}\right)=k$

So a radius scale of $a$ and a height scale of $b$ work whenever $a^2b=k$. For example, if $k=72$, then $a=6$ and $b=2$ would work because $6^2\cdot 2=72$.

Tags: cylinder volume, scale factor, radius squared