How to Find the Dimensions of a Cylinder If You Only Know the Volume?

Finding the unique dimensions (both radius and height) of a cylinder when you only know its volume is mathematically impossible. The volume of a cylinder is calculated using the formula:

$V = \pi \cdot r^2 \cdot h$

where $V$ is the volume, $r$ is the radius of the base, and $h$ is the height.

As per the provided reference:

• The formula is: Volume = $\pi \cdot \text{radius}^2 \cdot \text{height}$.

This equation has two unknown variables, $r$ and $h$, but only one equation (when only volume $V$ is known). You can find many different combinations of radius and height that result in the exact same volume.

What Information Do You Need?

To find both the radius and the height of a cylinder, in addition to the volume, you must know at least one other piece of information:

1. The height of the cylinder ($h$).
2. The radius of the cylinder ($r$).
3. A relationship between the radius and the height (e.g., the height is twice the radius).

Let's look at how to find one dimension if you know the other and the volume.

Finding Dimensions with Volume and One Other Value

You can rearrange the volume formula to solve for either the radius or the height.

1. Finding the Radius When You Know Volume and Height

If you know the volume ($V$) and the height ($h$), you can find the radius ($r$).

• Starting Formula: $V = \pi \cdot r^2 \cdot h$
• Rearrange to solve for $r^2$: $\frac{V}{\pi \cdot h} = r^2$
• Solve for $r$: $r = \sqrt{\frac{V}{\pi \cdot h}}$

Example (Based on Reference):

Suppose you have a cylinder with a volume of 791.28 cubic units and a height of 28 units. The reference provides steps similar to this:

1. Write the formula: Volume = $\pi \cdot \text{radius}^2 \cdot \text{height}$.
2. Substitute the dimensions: $791.28 = \pi \cdot r^2 \cdot 28$.
3. Solve the equation for the variable (using 3.14 for $\pi$ as in the reference):
   • $791.28 = 3.14 \cdot r^2 \cdot 28$
   • $791.28 = 87.92 \cdot r^2$
   • $r^2 = \frac{791.28}{87.92}$
   • $r^2 \approx 9$
   • $r \approx \sqrt{9}$
   • $r \approx 3$ units

Using this method, if you know the volume and height, you can successfully calculate the radius.

2. Finding the Height When You Know Volume and Radius

If you know the volume ($V$) and the radius ($r$), you can find the height ($h$).

• Starting Formula: $V = \pi \cdot r^2 \cdot h$
• Rearrange to solve for $h$: $\frac{V}{\pi \cdot r^2} = h$

Example:

Suppose you have a cylinder with a volume of 314 cubic units and a radius of 5 units. Using $\pi \approx 3.14$:

1. Start with the formula: $V = \pi \cdot r^2 \cdot h$
2. Substitute known values: $314 = 3.14 \cdot 5^2 \cdot h$
3. Simplify and solve for $h$:
   • $314 = 3.14 \cdot 25 \cdot h$
   • $314 = 78.5 \cdot h$
   • $h = \frac{314}{78.5}$
   • $h = 4$ units

Using this method, if you know the volume and radius, you can calculate the height.

In summary, while you cannot determine both the radius and height knowing only the volume, the cylinder's volume formula is essential for finding one dimension if the other is known.