# Volume of Cylinder

Ever wondered what shape a Pringles container looks like? Or how much sugar would be needed to fill it if it were emptied of all Pringles?

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Knowing what cylinders are and how to calculate their volume can easily aid you in measurements in reality because so many food items are stored in cylindrical containers.

## What is a cylinder?

A cylinder is a solid that has two identical circular flat ends connected with a tube.

A cylinder is seen in many daily use objects such as toilet tissue, candy container, tin milk container, pipes, etc.

### Types of cylinders

There are two basic types of cylinders.

The right circular cylinders: These cylinders have the planes of their bases perpendicular to the segment connecting the centers of the cylinder's circles.

An image of a right circular cylinder, StudySmarter Originals

The Oblique circular cylinder - These cylinders do not have the planes of their bases perpendicular to the segment connecting the centers of the cylinder's circles.

An image of an oblique circular cylinder, StudySmarter Originals

## How to calculate the volume of a cylinder?

### Volume of a circular cylinder

The volume of a circular cylinder is calculated by multiplying its height by the area of its circular base.

We recall that the area of a circle is given by,

$Are{a}_{circle}=\pi {r}^{2}$

Thus, the volume of a circular cylinder is given by,

$Volume{}_{circularcylinder}=Are{a}_{circularbase}×height=\pi {r}^{2}×h$

A cylindrical container has a base radius of 7 cm and a depth of 10 cm. Find the volume if $\mathrm{\pi }=\frac{22}{7}$

Solution:

We first note the radius and the height of the cylinder, $r=7cm,h=10cm$.

The volume of the circular cylinder is calculated as,

${V}_{circularcylinder}=\pi {r}^{2}×h=\frac{22}{7}×{7}^{2}×10=220×7=1540c{m}^{3}$

### Volume of an oblique circular cylinder

#### Cavalieri's Principle

Cavalieri's principle states that for any two solids having the same height and are such that their corresponding cross-sections at any level, have the same areas, then they have the same volume.

Cavalieri's principle is very important in finding volumes of oblique solid shapes. It enables us to use the same formula in calculating the volumes of these solids even though they are not straight.

According to Cavalieri's principle, considering two circular and oblique cylinders of the same height, having the same radius on their bases, we deduce that they will share the same cross-section areas. Hence, we can say that the volume of an oblique cylinder is equal to the volume of a right circular cylinder. Therefore the volume of an obliques cylinder, Vo is given by

${V}_{obliquecylinder}={V}_{circularcylinder}=\pi {r}^{2}×h$

Find the volume of the figure below, taking $\pi =\frac{22}{7}.$

Solution:

Recalling Cavalier's principle,

${V}_{obliquecylinder}={V}_{circularcylinder}=\pi {r}^{2}h$

We deduce from the figure that$r=9cm,h=28cm$.

Thus, the volume of the oblique cylinder given in the above figure can be calculated as,

${V}_{obliquecylinder}=\frac{22}{7}×{9}^{2}×28=22×81×4=7128c{m}^{3}$.

## What unit is the volume of a cylinder measured in?

The volume of a cylinder is measured in cubic centimeters cm3 and cubic meters m3 . Also, the volume of a cylinder is measured in liters l. Note that:

$1000c{m}^{3}=1l\phantom{\rule{0ex}{0ex}}1c{m}^{3}=0.001l\phantom{\rule{0ex}{0ex}}$

## Volume of a semicircular cylinder

A semicircular cylinder has its base and top as a semicircle. It is also known to be half of a right circular cylinder.

An image of a semicircular cylinder, StudySmarter Originals

The volume of a semicircular cylinder is calculated by dividing the volume of the completed cylinder by 2.

Imagine that the semicircular cylinder is completed to become a full cylinder. Thus,

$Volum{e}_{fullformedcylinder}=\pi {r}^{2}×h\phantom{\rule{0ex}{0ex}}$

Then the volume of a semicircular cylinder is given by,

${V}_{semicircularcylinder}=\frac{\pi {r}^{2}×h}{2}$

Find the volume of a semicircular cylinder with a height of 6 cm and a diameter of 5 cm. Take $\pi =\frac{22}{7}.$

Solution:

The volume of a semicircular cylinder is given by,

${V}_{semicircularcylinder}=\frac{\pi {r}^{2}×h}{2}$

We write down the height and the diameter from the given,$h=6cm,d=5cm$.

We deduce the radius from the diameter, $r=\frac{diameter}{2}=\frac{5}{2}cm.$

Hence, the volume of the semicircular cylinder is given by,

.

## How to calculate the volume of irregular shapes?

Knowledge of the volume of regular solids makes the calculation of irregular shapes possible. Firstly, you have to break down the irregular solid to its regular solid components then you determine its volume.

Let's see how this can be done in the following example.

Determine the volume of the casket below. Take $\pi =\frac{22}{7}.$

Solution:

We first note that the top of the casket is a semicircular cylinder while the base is a rectangular prism.

Let us find the volume of the semicircular cylindrical top.

${V}_{semicircularcylinder}=\frac{\pi {r}^{2}×h}{2}$

We note that the diameter of the semicircular cylinder is $d=14cm.$Thus, $r=\frac{diameter}{2}=\frac{d}{2}=\frac{14}{2}=7cm.$

Hence,

.

The volume of the rectangular prism,

${V}_{rec\mathrm{tan}gularprism}=length×breadth×heightoftheprism$

From the figure, we deduce that length = 30 cm, breadth = 14 cm and height = 15 cm.

Hence,

${V}_{rec\mathrm{tan}gularprism}=30×14×15=6300c{m}^{3}.$

The volume of the casket is calculated as the sum of the volume of the semicircular cylinder and the volume of the rectangular prism.

${V}_{casket}={V}_{semicircularcylinder}+{V}_{rec\mathrm{tan}gularprism}=2310+6300=8610c{m}^{3}$.

How many tissue rolls does Brenda need to block 40 425 cubic centimeters opening in her room if the height of the roll is 50 cm? Take $\pi =\frac{22}{7}.$

Solution:

To determine how many rolls of tissues Brenda has to use, we need to find the volume of the tissue, ${V}_{tissue}$.

The volume of the tissue can be calculated by subtracting the volume of the tissue's hollow space, from the volume of the whole cylinder.

Thus,

${V}_{tissue}={V}_{wholecylinder}-{V}_{hollowspace}$

We calculate first the volume of the whole cylinder,

${V}_{wholecylinder}=\mathrm{\pi }×{\mathrm{r}}^{2}×\mathrm{h}=\mathrm{\pi }×{\left(\frac{28}{2}\right)}^{2}×50=\frac{22}{7}×{14}^{2}×50=30800{\mathrm{cm}}^{3}$

Next, in order to calculate the volume of the hollow space, we first need to calculate its corresponding radius. But the diameter of the hollow space can be found by subtracting the diameter of the whole cylinder from the diameter of the non-empty cylinder, thus

$diamete{r}_{hollowcylinder}=28-7=21cm$

Now, the volume of the hollow space is,

${V}_{hollowspace}=\pi ×{r}^{2}×h=\frac{22}{7}×{\left(\frac{21}{2}\right)}^{2}×50=17325{\mathrm{cm}}^{3}.$

Thus the volume of the tissue is,

${V}_{tissue}={V}_{wholecylinder}-{V}_{hollowspace}=30800-17325=13475c{m}^{3}.$

Since the volume of the space Brenda is to fill is 40 425 cm3, then she would need,

$\left(40425÷13475\right)tissues=3tissues$.

## Volume of Cylinder - Key takeaways

• A cylinder is a solid that has two identical circular flat ends connected with a tube.
• The two types of cylinders are the right circular and oblique circular cylinders.
• Cavalieri's principle states that for any two solids which possess the same height as well as cross-sectional area, their volumes are the same.
• The volume of a cylinder is given by ${V}_{cylinder}=\pi ×{r}^{2}×h.$
• A semicircular cylinder has its base and top as a semicircle. It is also known to be half of a right circular cylinder.

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##### Frequently Asked Questions about Volume of Cylinder

Find the volume of a cylinder.

The volume of a cylinder is calculated by multiplying the area of its circular base by the height of the cylinder.

What is the formula for finding the volume of a cylinder ?

The formula for finding the volume of a cylinder is; pie times the square of radius times the height.

What is the volume of the right cylinder?

The volume of a right cylinder is calculated in the same way as calculating the volume of a cylinder.

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