Last updated: Apr 17, 2024

Hollow Cylinder Volume Calculator

Q: What is the volume of a hollow cylinder with D = 8, d = 4, and h = 12?

The volume is 452.5 cm³ . To find the volume of a hollow cylinder with diameters D = 8 cm and d = 4 cm , and height h = 12 cm , follow these steps: Calculate the radii: R = D/2 = 4 cm , and r = d/2 = 2 cm . Calculate the volume of a "crown" with such measurements: π × (R² - r²) = π × (4² -2²) = 37.7 cm² Multiply this quantity by the height of the cylinders to find the volume: VH = 37.7 × 12 = 452.4 cm³ Read more

Table of contents

What is a hollow cylinder?How to calculate the volume of a hollow cylinder (with thickness and without)Examples of applications of the volume of hollow cylinder formula FAQs

Bring some math into your geometry with our hollow cylinder volume calculator: With this simple tool, you will learn how to calculate the difference between cylinders and not only! Keep reading to find:

What is a hollow cylinder?
How to find the volume of a hollow cylinder;
The formula for the volume of a hollow cylinder with thickness; and
Exemplary calculations for the volume of such solid.

What is a hollow cylinder?

A hollow cylinder is an interesting three-dimensional shape obtained from the subtraction of two cylinders that share the central axis.

You can find hollow cylinders in many, many places! Pipes, bottles, thick jars, and so on. Most of the time, the hollow cylinders we meet are bent or flexible: in this case, straighten them before computing their volume!

How to calculate the volume of a hollow cylinder (with thickness and without)

To calculate the volume of a hollow cylinder, you can use a rough but effective formula and subtract the volume of the two right cylinders:

V_{\text{H}} = V_{\text{C}_1}-V_{\text{C}_2}

where:

$V_{\text{H}}$ — Volume of the hollow cylinder;
$V_{\text{C}_1}$ — Volume of the bigger, outer cylinder; and
$V_{\text{C}_2}$ — Volume of the smaller, inner cylinder.

However, if we expand the formula for the volume of a cylinder, the formula for the volume of the hollow cylinder becomes:

\!\begin{align*} V_{\text{H}} &= V_{\text{C}_1}-V_{\text{C}_2}\\[.5em] &=\left(\left(\pi\cdot R^2\right)\!\cdot\! h \right)\!-\!\left(\left(\pi\cdot r^2\right)\!\cdot\! h \right)\\[.5em] &=\pi\cdot\left(R^2-r^2\right)\cdot h, \end{align*}

where:

$h$ — Height of both cylinders;
$R$ — Radius of the larger cylinder; and
$r$ — Radius of the inner cylinder.

In this equation for the volume of a hollow cylinder, you can also almost identify the thickness of the hollow cylinder:

t = R-r

However, there is a square: don't fall for the mistake of comparing $R^2-r^2$ with $R-r$ .

If you want to learn how to find the volume of a hollow cylinder with its thickness, this formula is what you need:

V_{\text{H}}=\pi\cdot\left(t^2-2\!\cdot\! t\!\cdot\! r\right)\cdot h

As you can see, in the equation for the volume of a hollow cylinder with thickness, we still have to specify the inner radius: in hindsight, it makes sense!

You can also replace $R$ and $r$ with $D/2$ and $d/2$ , respectively, where $D$ is the diameter of the larger cylinder, and $d$ is the diameter of the smaller or inner cylinder to have the following equation:

\!\begin{align*} V_{\text{H}} &=\pi\cdot\left(R^2-r^2\right)\cdot h\\[.5em] &=\pi\cdot\left(\left(\frac{D}{2} \right)^2-\left(\frac{d}{2}\right)^2\right)\cdot h\\[1.5em] &=\pi\cdot\left(\frac{D^2 - d^2}{4}\right)\cdot h \end{align*}

🙋 This equation is what we mainly use in our hollow cylinder calculator. But you can tick the Show outer and inner radii checkbox of our tool to enter your hollow cylinder's outer and inner radii measurements.

If you want to calculate the inner volume of a hollow cylinder and not its shell, visit our pipe volume calculator: don't worry, the math is similar!

And for other cylinder calculators, visit the radius of a cylinder calculator, the height of a cylinder calculator, or the surface area of a cylinder calculator

Examples of applications of the volume of hollow cylinder formula

Arthur C. Clarke wrote an entire book about a hollow cylinder: Encounter with Rama. In the novel, a group of astronauts meets a gargantuan alien spaceship in the shape of an almost featureless hollow cylinder. Rama, the spaceship, is $50\ \text{km}$ long and has an outer diameter of $16\ \text{km}$ (megastructures play important roles in science-fiction). The thickness of the spaceship is $2\ \text{km}$ . What's the volume of the "shell" of Rama?

You know how to calculate the volume of a hollow cylinder: apply the formula of the hollow cylinder volume with thickness after calculating the inner diameter:

$r = R-t=(8-2)\ \text{km}=6\ \text{km}$

Now you have all the quantities needed to calculate the volume:

\begin{align*} V_{\text{H}}&=\pi\cdot\left(t^2-2\!\cdot\! t\!\cdot\! r\right)\cdot h\\[.5em] &=\left(\pi\cdot\left(2^2-2\!\cdot\!2\!\cdot\!6\right)\cdot 50\right) \text{km}^3\\[.5em] & = 2,\!356.19\ \text{km}^3 \end{align*}

The volume of Rama's shell is huge: you can fit more than two million Empire States Buildings in that space. It is no surprise that futuristic projects for interstellar ships involve such structures you can fit many people in such volumes!

Our volume of a hollow cylinder calculator simplifies your tasks: you only have to input the data. Thickness, inner, and outer diameter are already linked in our tool: if you give us two of them, we will calculate the third!

FAQs

How do I calculate the volume of a hollow cylinder?

To calculate the volume of a hollow cylinder, you can either subtract the volumes of the two cylinders that create the shape or use the following formula:

VH = V₁ − V₂ = π × (R² - r²) × h

where:

VH is the volume of the hollow cylinder, and V₁ and V₂ are the cylinders' volumes;
R and r the radii of the cylinders; and
h their height.

What is the volume of a hollow cylinder with D = 8, d = 4, and h = 12?

The volume is 452.5 cm³. To find the volume of a hollow cylinder with diameters D = 8 cm and d = 4 cm, and height h = 12 cm, follow these steps:

Calculate the radii: R = D/2 = 4 cm, and r = d/2 = 2 cm.
Calculate the volume of a "crown" with such measurements:

π × (R² - r²) = π × (4² -2²) = 37.7 cm²
Multiply this quantity by the height of the cylinders to find the volume:

VH = 37.7 × 12 = 452.4 cm³

How do I calculate the thickness of a hollow cylinder?

To calculate the thickness of a hollow cylinder, subtract the inner and outer cylinder radii, r and R: the thickness of the solid is:

t = R − r

You can calculate the volume of a hollow cylinder using thickness with the formula:

VH = π × (t² + 2tr²) × h

Can I calculate the volume of an asymmetric hollow cylinder?

Yes! Since the volume of a cylindrical shell equals the difference between the volumes of the two cylinders from which we build the solid itself, it doesn't matter if these two are aligned or not till they share the same height and angle.