# Tank Volume Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Jan 04, 2023

With this tank volume calculator, you can easily estimate the volume of your container. Choose between nine different tank shapes: from standard rectangular and cylindrical tanks to capsule and elliptical tanks. You can even find the volume of a frustum in cone bottom tanks. Just enter the dimensions of your container, and this tool will calculate the total tank volume for you. You may also provide the fill height, which will be used to find the filled volume.

Do you wonder how it does it? Scroll down, and you'll find all the formulas you need – the volume of a capsule tank, elliptical tank, or the widely-used cone bottom tanks (sometimes called conical tanks), as well as many more!

Are you looking for other types of tanks in different shapes and for other applications? Check out our volume calculator to find the volume of the most common three-dimensional solids. For something more specialized, you can also have a glance at the aquarium calculator and pool volume calculators for solutions to everyday volume problems.

## How to use the tank volume calculator

This tank volume calculator is a simple tool that helps you find the volume of the tank as well as the volume of the filled part. You can choose between ten tank shapes:

• Vertical cylinder;
• Horizontal cylinder;
• Rectangular prism (box);
• Vertical capsule;
• Horizontal capsule;
• Vertical oval (elliptical);
• Horizontal oval (elliptical);
• Cone bottom;
• Cone top; and
• Frustum (truncated cone, funnel-shaped)

"But how do I use this tank volume calculator?", you may be asking. Let's have a look at a simple example:

1. Decide on the shape. Let's assume that we want to find the volume of a vertical capsule tank – choose that option from the drop-down list. The schematic picture of the tank will appear below; make sure it's the one you want!

2. Enter the tank dimensions. In our case, we need to type in the length and diameter. In our example, they are equal to $30\ \mathrm{in}$ and $24\ \mathrm{in}$, respectively. Additionally, we can enter the fill height – $32\ \mathrm{in}$.

3. The tank volume calculator has already found the total and filled volume! The total volume of the capsule is $90.09\ \mathrm{US\ gal}$, and the volume of the liquid inside is $54.84\ \mathrm{US gal}$. As always, you can change the units by clicking on the volume units themselves. Easy-peasy!

## Cylindrical tank volume formula

To calculate the total volume of a cylindrical tank, all we need to know is the cylinder diameter (or radius) and the cylinder height (which may be called length if it's lying horizontally).

Vertical cylinder tank

We may find the total volume of a cylindrical tank with the standard formula for volume – the area of the base multiplied by the height. A circle is the shape of the base, so its area, according to the well-known equation, is equal to $\pi \times \mathrm{radius}^2$. Therefore the formula for a vertical cylinder tanks volume looks like this:

$\begin{split} V_{\scriptsize{\mathrm{\tiny vertical\ cylinder}}} &= \pi \times r^2 \times h\\[.6em] &=\pi\times\left(\frac{d}{2}\right)2\times h, \end{split}$

where:

• $r$Radius of the base; and
• $h$ — Height of the cylinder.

If we want to calculate the filled volume, we need to find the volume of a "shorter" cylinder – it's that easy!

$\begin{split} V_{\scriptsize{\mathrm{\tiny vertical\ cylinder}}} &= \pi \times r^2 \times f\\[.6em] &=\pi\times\left(\frac{d}{2}\right)2\times f, \end{split}$

where $f$ is the height of the filled portion of the cylinder.

Horizontal cylinder tank

The total volume of a horizontal cylindrical tank may be found in an analogical way – it's the area of the circular end times the length of the cylinder:

$\begin{split} V_{\scriptsize{\mathrm{\tiny horizontal\ cylinder}}} &= \pi \times r^2 \times l\\[.6em] &=\pi\times\left(\frac{d}{2}\right)2\times l, \end{split}$

where $l$ is the length of the cylinder.

Things get more complicated when we want to find the volume of the partially filled horizontal cylinder. First, we need to find the base area: the area of the circular segment covered by the liquid:

$A_{\mathrm{\tiny segment}} = \frac{1}{2}\times r^2 \times (\theta-\sin(\theta)),$

where $r$ is the radius of the base, and $\theta$ is the central angle of the segment. The angle $\theta$ may be found from the formula for cosine:

$\cos\left(\frac{\theta}{2}\right) = \frac{r-f}{f},$

where $f$ is the height of the filled section of the base.

Therefore:

$\theta= 2\times \arccos\left(\frac{r-f}{f}\right)$

And finally, the formula for the volume of a partially filled horizontal cylinder $V_{\scriptsize{\mathrm{fhc}}}$ is:

$\small V_{\scriptsize{\mathrm{fhc}}} = 0.5\times r^2\times(\theta -\sin(\theta))\times l,$

where:

$\theta = 2\times \arccos\left(\frac{r-f}{r}\right)$

If the cylinder is more than half full, then it's easier to subtract the empty tank part from the total volume.

## Rectangular tank volume calculator (rectangular prism)

If you're wondering how to calculate the volume of a rectangular tank (also known as cuboid, box, or rectangular hexahedron), look no further! You may know this tank as a rectangular tank – but that is not its proper name, as a rectangle is a 2D shape, so it doesn't have a volume.

To find the rectangular prism volume, multiply all the dimensions of the tank:

$V_{{\mathrm{rectangular\ prism}}} = h\times w\times l,$

where:

• $h$ — The height of the tank;
• $w$ — The width; and
• $l$ — The length of the tank.

If you want to know what the volume of the liquid in a tank is, simply change the height variable into filled in the rectangular tank volume formula. The volume of a rectangular filled prism, $V_{\scriptsize{\mathrm{rpf}}}$, is:

$V_{{\mathrm{rpf}}} = f\times w\times l,$

where $f$ is the height of the filled portion.

For this tank volume calculator, it doesn't matter if the tank is in a horizontal or vertical position. Just make sure that filled and height are along the same axis.

## Formula for volume of a capsule

Our tool defines a capsule as two hemispheres separated by a cylinder. To calculate the total volume of a capsule, all you need to do is add the volume of the sphere to the cylinder part:

$V_{\scriptsize{\mathrm{cap}}} = \pi\!\times\!\! \left(\frac{d}{2}\right)^2\!\!\!\times\!\left(\frac{4}{3}\!\times\!\frac{d}{2}\!+\!l\right)$

Depending on the position of the tank, the filled volume calculations will differ a bit:

1. For horizontal capsule tank

As the hemispheres on either end of the tank are identical, they form – add this part $V_{\scriptsize{\mathrm{sph\ hf}}}$ to the part from the horizontal cylinder $V_{\scriptsize{\mathrm{fhc}}}$ (check the paragraph above) to calculate the volume of the liquid in a filled horizontal capsule ($V_{\scriptsize{\mathrm{cap\ hf}}}$):

$\begin{split} &V_{\scriptsize{\mathrm{cap\ hf}}} = V_{\scriptsize{\mathrm{fhc}}}+V_{\scriptsize{\mathrm{sph\ hf}}}\\[.6em] &= \frac{1}{2}\!\times\!\! \left(\frac{d}{2}\right)^2\!\!\! \times\! (\theta-\sin(\theta))\times l \\[1em] &\quad+\frac{\pi\times f^2}{3}\!\times\!\left(\frac{3}{2}\times d-f\right), \end{split}$

where:

• $d$ — Diameter of the tank;
• $f$ — Filled height; and
• $\theta$ — The angle corresponding to the filled sector in the cylinder tank.

2. For vertical capsule tank

The formula differs for various fill heights. In all the next formulas, we will meet these quantities:

• $d$Diameter of the tank;

• $f$ — Height reached by the liquid; and

• $l$Length of the cylindrical section.

• If $f < d/2$, then the liquid is only in the bottom hemisphere part, so we only need the volume of a spherical cap formula:
$V_{\scriptsize{\mathrm{cap\ f}}} = \frac{(\pi\!\times \!f)^2}{3}\!\times\!\left(\frac{3}{2}\!\times \!d - f\right)$
• If $d/2 < f < d/2 +l$, then we need to add the hemisphere volume and "shorter" cylinder:
$\quad\ \ \ \begin{split} &V_{\scriptsize{\mathrm{cap\ f}}} = \frac{2}{3}\times \pi\left(\frac{d}{2}\right)^3\\[1em] &\qquad+\pi\times\left(\frac{d}{2}\right)^2\times\frac{f-d}{2} \end{split}$
• If $d/2 + l < f$, it means that we have a full bottom hemisphere and cylinder, so we just need to subtract the spherical cap (empty part) from the whole volume:
$\begin{split} &V_{\scriptsize{\mathrm{cap\ f}}} = V_{\scriptsize{\mathrm{cap}}}\! -\! \frac{\pi\times(l+d-f)^2}{3}\\ &\qquad\ \ \times\left(\frac{3}{2}\times d - (l+d-f)\right) \end{split}$

## Elliptical tank volume (oval tank)

In our calculator, we define an oval tank as a cylindrical tank with an elliptical end (not in the shape of a stadium, as it is sometimes defined). To find the total volume of an elliptical tank, you need to multiply the ellipsis area times the length of the tank:

$V_{\scriptsize{\mathrm{ellipse}}} = \pi\!\times\!w\!\times\!l\!\times\!\frac{h}{4},$

where:

• $w$Width of the tank;
• $h$Height of the tank; and
• $l$ — The tank's length.

Finally, another easy formula! Unfortunately, finding the volume of a partially filled tank – both in the horizontal and vertical positions – is not so straightforward. You need to use the formula for the ellipse segment area and multiply the result times the length of the tank:

$\begin{split} &V_{\scriptsize{\mathrm{ell\ f}}} = l\!\times\!h\!\times\!\frac{w}{4}\\[1em] &\qquad\times\!\left[\arccos\left(1\!-\!\frac{2\!\times\!f}{h}\right)\right.\\[1.5em] &\!\!\left.-\!\left(\!1\!-\!\frac{2\!\times\!f}{h}\!\right)\!\times\!\sqrt{\frac{4\!\times\!f}{h}\!-\!\frac{4\!\times\!f^2}{h^2}}\right] \end{split}$

## Frustum volume – tank in the shape of a truncated cone

To calculate the truncated cone volume (a particular shape we met at our truncated cone calculator), use the formula:

$\begin{split} &V_{\scriptsize{\mathrm{frustum}}} = \frac{1}{3}\times\pi\times h_{\mathrm{cone}}\\[1em] &\qquad\times\left( \left(\frac{d_{\mathrm{top}}}{2}\right)^2\!+\! \frac{d_{\mathrm{top}}}{2}\!\times\!\frac{d_{\mathrm{bot}}}{2}\!\right.\\[1em] &\qquad\qquad \left.+\!\left(\frac{d_{\mathrm{bot}}}{2}\right)^2\right), \end{split}$

where:

• $d_{\mathrm{top}}$Upper diameter of the frustum;
• $d_{\mathrm{bot}}$Lower diameter of the frustum;
• $h_{\mathrm{cone}}$Height of the tanks.

If you want to find the partially filled frustum volume for a given fill height, calculate the top radius of the filled part first:

$R = \frac{1}{2} \times d_{\mathrm{top}}\times \frac{f+z}{h_{\mathrm{cone}}+z},$

where:

$z = h_\mathrm{cone} \times \frac{d_{\mathrm{bot}}}{d_{\mathrm{top}}-d_{\mathrm{bot}}}$

(You can derive the formula from triangles' similarity.)

Afterward, just find the new frustum volume:

$\begin{split} &V_{\scriptsize{\mathrm{ff}}}=\frac{1}{3} \times \pi \times f \\ &\ \ \times \!\left(R^2 \!+ \!R\! \times \!\frac{d_{\mathrm{bot}}}{2} + \left(\frac{d_{\mathrm{bot}}}{2}\right)^2\right) \end{split}$