# Volume of a Cylinder

Shapes

We’re all surrounded by many different shapes! Right from your television, your clock, your mug – shapes are all around us. They come in different sizes and have certain attributes (or physical features) that distinguish them from other shapes. For example, a square (like a chess board, for example) has all four sides being of equal length, and all at right angles to each other. Likewise, a rectangle (like a door, for example) has opposite sides being of equal length! In this way, we can distinguish between shapes – by observing the differences in their physical attributes.

As mentioned before, we are surrounded by several different shapes. One of these shapes is known as the cylinder! You can see cylinders in cans, mugs, gas cylinders, etc. Let’s learn more about this unique shape!

## What are Cylinders?

Have you ever observed the shape of a soda can? It is composed of 2 flat circles that are parallel to each other, joined together by a curved surface. Such a shape is known as a cylinder.

Examples of Cylinders in Daily Life
Look around you! You are surrounded by so many different shapes. Some of these could also be found in your house, and they are all examples of cylinders!

• Soda Cans
• Candles
• Gas Cylinders
• Drums
• Batteries

## Components of a Cylinder

As mentioned before, a cylinder is made up of 2 flat circles that are connected by a curved surface. Aside from this, a cylinder also has-

Radius of the Cylinder – the radius of either of the two identical circles that the cylinder is made of provides the radius for the entirety of the cylinder.
Height of the cylinder – the height of the cylinder is the perpendicular distance between the two parallel circular bases.

Fun fact: If the curved surface of a cylinder is flattened out, it forms a new shape – the rectangle! The diagram below explains this more clearly-

## Types of Cylinders

There are several types of cylinders, which all look slightly different from one another, but all fall under the broad category of cylinders.

Four common types of cylinders are –

Right Circular Cylinder – In the case of this type of cylinder, two flat circular bases are joined together by a circular base in such a way that if an imaginary line was drawn connecting the centers of both circles, they would be perpendicular to the circular bases. Example – Pipes
Elliptic Cylinder – In simple words, an elliptic cylinder is such that instead of a circular base, it is in an elliptical shape. Example – Optical Lenses
Oblique Cylinder – An oblique cylinder is when the two circular bases are parallel to each other, but the sides are not perpendicular, but are in fact, tilted. An oblique cylinder is also called a slanted cylinder. Example – The Leaning Tower of Pisa.
Right Circular Hollow Cylinder/Cylindrical Shell – Cylindrical shells are seen when a cylinder is contained within another cylinder. Both these cylinders are right circular cylinders. The important property of this type of cylinder is that it is hollow (empty) on the inside. Example – Hydraulic Cylinders.

Of these, the most commonly used type of cylinder is the right circular cylinder.

## Properties of a Cylinder

Faces of a Cylinder – The face of any object refers to a single curved or flat surface on said object. For example, a cylinder has 3 faces – two faces being of the two flat surfaces of the circular ends, and one curved surface that connects these two circles.

Vertices of a Cylinder – The vertex of any object is the point where any 2 faces on said object meet. For example, in the case of a cylinder, the shape has no vertices, as none of the faces are actually coinciding with each other.

Edges of a Cylinder – An edge refers to any line segment that joins 2 vertices. For example, a cylinder has 2 edges – the 2 edges being the 2 perpendicular line segments that join the 2 points on the circumference of each of the 2 circles.

## Curved Surface Area of an Object

Imagine an ice cream cone. When the salesman/saleswoman hands you the cone, you would have seen them provide you with a tissue around the curved surface of the cone.

How do you think they know how much tissue to provide you with, so that the entirety of the cone’s curved area is covered by the tissue?

This can be found by calculating the cone’s curved surface area. The curved surface area of any object – as the name suggests – is the area of only the curved surface of an object.

Note: This value can only be calculated if the 3D object contains at least one curved surface.

For example, in the case of a cube – the shape contains no curved surface, and hence will not have a curved surface area (CSA). However, a shape like a sphere is bound to have a curved surface area, as it is a shape that is entirely composed of a curve!

### Curved Surface Area of a Cylinder

The CSA of a cylinder can be found by the formula 2πr x h.

### Derivation of the CSA of a Cylinder

You may be wondering how we reached the formula of 2πr x h.

Let’s solve it together!

As you are already aware, a cylinder has 2 identical and parallel circles that are joined together by a curved surface – the perpendicular distance of which is known as the height.

The circumference of a circle is found by the formula 2πr. Since the two circles are identical, we do not need to add the circumferences of both these circles to our equation as their circumferences would be the same.
However, there is still one more curved surface surrounding the cylinder! We thus multiply the value of the curved surface (h) with our value for 2πr.

We are thus left with the formula 2πrh, which is the formula used to find the CSA of a cylinder!

### Area of a Circle

The area of a circle is important when considering that a cylinder is flanked by two circles on both ends.

The area of a circle can be found simply by the formula 2πr.

However, since a cylinder has 2 circles, we have to take into account the areas for both these circles. Both these circles are identical, and hence have the same radius. Thus, the areas for both these circles can be found by multiplying the value of ‘r’ for both these circles, as they will have the same radius. Therefore, the new formula for the area of both the circles in a cylinder would just be 2πr2.

### Total Surface Area (TSA) of an Object

Imagine yourself holding a small wooden box shaped like a cube. You want to paint the box in such a way that every inch of the box is covered. However, you are unsure of how much paint you require.

How can you find the exact amount of paint needed to cover the entire box?

The answer is by finding the total surface area of the cube. The total surface area of any object refers to the area of all the exterior faces of any 3D object. The unit we use to measure the area of an object is cm2, and for larger objects, we use m2.

Total Surface Area (TSA) of a Cylinder
The surface area of a cylinder is essentially the sum of the area of the curved surface of the cylinder and the area of the 2 circular bases of the cylinder. Thus, the surface area of a cylinder is found by the formula – (2π × r × h) + (2πr2) = 2πr (h + r) Square units.

Volume of an Object
Observe any hollow cylinder around you. Fill water up to the cylinder’s brim. You will notice that the cylinder can only contain a certain amount of water before it overflows from the container. This constant amount of water that can be filled inside that cylinder is dependent on the volume of the cylinder.

Now, you may ask, what exactly do we mean by volume?

Volume, in very simple terms, is the amount of space contained within an object. The unit by which we measure volume of an object is the cm3. For objects that are larger in size, we use m3.

Note: The volume of an object can only be found if the object is 3-dimensional (3D). This is because a 2-dimensional object would not contain any value for ‘height’ – which is necessary when calculating the volume of an object.

## Volume of a Cylinder

The volume of a cylinder is represented by the formula – πr2h, where r represents the radius of the cylinder, and h represents the height of the cylinder.

Derivation of the Volume of a Cylinder
As discussed previously, the volume of any substance is simply the space occupied within said object. So if we observe a cylinder, we already know that it is a 3D object consisting of a curved base flanked on both ends by two parallel circles.

The surface area of a circle is πr2. Since there are two circles, it would be natural to assume that the total surface area of both circles combined would be 2πr2 . However, it is important to remember that both circles are identical, and thus, their surface areas would be the same, that is, πr2. If we multiply this value by the height (h) of the cylinder, we gain the formula πr2h, which is the formula of the volume of a cylinder.

Summary
Now that we have learnt so much about cylinders, let’s make note of everything that we have learnt in this lesson –

A cylinder has two flat circular bases on both ends, joined by a curved base.
A cylinder has a radius and a height.
A flattened cylinder forms a rectangle.
There are four types of cylinders – oblique cylinder, right circular cylinder, elliptical cylinder, and right circular hollow cylinder.
The curved surface area of an object is the area of only the curved part of a 3D object.
The curved surface area of a cylinder (CSA) is 2πr x h.
The total surface area of an object (TSA) is the surface area of all exterior faces of an object.
The TSA of a cylinder is 2πr (h + r).
The volume of an object is the amount of space contained within it.
The volume of a cylinder is πr2h.

Quick Questions!

Now that we’ve learnt so much about cylinders, let’s solve some quick mental-math questions together! Take a look at the summary above before attempting these questions!

1) What is the TSA of a cylinder where the radius measures 5cm, and the height measures 13cm?
A) 100.89 cm2
B) 565.48 cm2
C) 408.40 cm2
D) 661.29 cm2

2) If the height of a cylinder is 10cm, and the volume of the cylinder is 465 cm2, what is the radius of the cylinder?
A) 5.69 cm
B) 2.98 cm
C) 4.33 cm
D) 3.84 cm

3) Which example would require the knowledge of the area of the shape?
A) Figuring out how many marbles can completely fit in a glass jar.
B) Figuring out the maximum capacity of a water tank.
C) Figuring out how much petrol can be pumped into a car.
D) Figuring out how much wire is needed to surround the perimeter of a park.

4) What is the radius of a circle if the area is 200cm2?
A) 9.89 cm
B) 6.35 cm
C) 7.97 cm
D) 8.63 cm

5) A scientist is conducting an experiment. She wants to know what is the maximum amount of water that can be filled in a cylindrical vase without it overflowing. The cylinder has a radius of 6cm and a perpendicular height of 9 cm.
A) 1017.87 cm3
B) 1017.87 cm2
C) 565.46 cm3
D) 565.46 cm2

6) You are given a cylinder, and asked to figure out the curved surface area of the cylinder. The known values are: 15 cm radius and 25 cm height.
A) 3560.83 cm2
B) 880.25 cm 2
C) 2840.67 cm2
D) 2356.19 cm2

Solution:
1) B
2) D
3) D
4) B
5) A
6) D

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