Volume of a Cylinder

Cylinders are solid shapes found in many everyday objects. It is a three-dimensional solid with the most basic of curvilinear shapes. From water bottles to gas tanks, they are all around us. Understanding the volume of a cylinder helps in various practical situations like storing liquids, calculating space, or solving math problems.

1.0What Is a Cylinder?

A cylinder is a three-dimensional shape. It has two parallel circular bases and a curved surface that connects them. The distance between the two bases is called the height of the cylinder.

Key parts of a cylinder:

• Radius (r): The radius of the circular base.
• Height (h): The distance between the two circular bases.

2.0Formula for Volume of a Cylinder

The volume of a cylinder means how much space it occupies. It tells us how much liquid or material it can hold.

The formula for volume of a cylinder is:

$V=\pi r^{2}h$

Where:

• V is the volume,
• R is the radius of the base,
• h is the height,
• π ≈ 3.14159

3.0Volume of a Cylinder in Litres

Sometimes, we need the volume of a cylinder in litres. For this, we convert the volume from cubic centimetres to litres.

• 1 litre = 1000 cubic centimetres (cm³)

If you calculate the volume in cm³, divide the result by 1000 to get the volume in litres.

Example:

If the volume is 5000 cm³:

$\text{Volume in litres}=\frac{5000}{1000}=\text{litres}$

4.0Volume of a Cylinder Derivation

Let’s look at the volume of a cylinder derivation.

We know the area of a circle is: 

$Area=\pi r^2$

A cylinder is formed by stacking circles on top of each other along its height.

So, multiplying the area of the base by the height gives the volume:

$\text{Volume = area of base}\times height=\pi r^{2}h$

That’s the volume of a cylinder derivation.

5.0Surface Area and Volume of a Cylinder

Along with volume, the surface area and volume of a cylinder are also important.

There are two types of surface areas:

• Lateral surface area (LSA): The area around the side.
• Total surface area (TSA): The area including the top and bottom circles.

Formulas

• Lateral surface area: $2\pi rh$
• Total surface area: $2\pi r(h+r)$
• Volume: $\pi r^{2}h$

6.0How to Find the Volume of a Cylinder

To find the volume of a cylinder, follow these steps:

1. Measure the radius of the circular base.
2. Measure the height of the cylinder.
3. Use the formula $V=\pi r^{2}h$
4. Plug in the values and calculate.

Example 1

Find the volume of a cylinder with:

• Radius = 5 cm
• Height = 10 cm

Solution: 

$$\begin{gathered} V=\pi r^{2}h=\pi \times 5^2\times 10=250\pi \\ V\approx 250\times 3.1416=785.4{\text{cm}}^3 \end{gathered}$$

Example 2

Find the volume of a cylinder with:

• Radius = 7 cm
• Height = 15 cm

Solution: 

$$\begin{gathered} V=\pi r^{2}h=\pi \times 7^2\times 15=735\pi \\ V\approx 735\times 3.1416=2309.1{\text{cm}}^3 \end{gathered}$$

7.0Questions on Volume of a Cylinder

Here are a few questions on volume of a cylinder to practice:

Question 1

A cylinder has a radius of 3 cm and a height of 12 cm. Find the volume.

Solution: 

$$\begin{gathered} V=\pi r^{2}h=\pi \times 3^2\times 12=108\pi \\ V\approx 339.29{\text{cm}}^3 \end{gathered}$$

Question 2

A cylindrical water tank has a radius of 2 m and a height of 3.5 m. Find the volume in litres.

Solution: 

$$\begin{gathered} V=\pi r^{2}h=\pi \times 2^2\times 3.5=14\pi \\ V\approx 43.98{\text{m}}^3 \end{gathered}$$

Now convert to litres:

1 m³ = 1000 litres

43.98 x 1000 = 43980 liters.

Question 3

Find the height of a cylinder if:

Radius = 6 cm

Volume = 2261.95 cm³

Solution: 

$$\begin{gathered} V=\pi r^{2}h\Rightarrow 2261.95=\pi \times 6^2\times h \\ h=\frac{2261.95}{\pi \times 36}\approx 20\text{cm.} \end{gathered}$$

Question 4

A cylindrical container has a diameter of 10 cm and a height of 20 cm. Find the volume in cm³ and litres.

Solution: 

$$\begin{gathered} \text{First, find the radius:}Radius=\frac{10}{2}=5\text{cm.} \\ \text{Use the formula: } \\ V=\pi r^{2}h=\pi \times 5^2\times 20=500\pi \\ V\approx 500\times 3.1416=1570.8{\text{cm}}^3 \\ \text{Convert to litres:} \\ \frac{1570.8}{1000}=1.5708\text{liters} \end{gathered}$$

Question 5

The volume of a cylinder is 3141.6 cm³. If the radius is 10 cm, find the height.

Solution: 

$$\begin{gathered} V=\pi r^{2}h\Rightarrow 3141.6=\pi \times 10^2\times h \\ h=\frac{3141.6}{314.16}=10\text{cm.} \end{gathered}$$