Let us carry out a small activity. Take a coin. We know that coin is circular in shape. Now place another coin on the first coin and so on. You will see that when you place such coins, the structure that you get is a cylinder, which is three dimensional. Let us learn about the volume of a Cylinder in this chapter.

**Download Cylinder Formula Cheat Sheet Below**

## Cylinder

A cylinder is a three-dimensional object with two round flat bases and one curved side. It has a curved surface in the middle. The base and the top surface are identical. That means the bases are always parallel and congruent to each other. It has no vertices.

**Browse more Topics under Mensuration**

- Circles
- Polygons
- Rectangles and Squares
- Trapezium, Parallelogram and Rhombus
- Area and Perimeter
- Cube and Cuboid

### Total Surface Area of a Cylinder

Now if we look at this figure carefully, we see that there three faces of the cylinder. Two circles and one rectangle. One circle is at the base of the cylinder and other is at the top. Both of these circles are of the same size. Rectangle face is the curved surface of the cylinder. So,

- The area of the circle of the cylinder is πr².
- The area of two circles will be 2πr².
- The radius ‘r’ of a cylinder is the radius of its base. Now, the area of the rectangle = length × breadth. 2πr is the circumference of the circle and h is the height. Area of the curved surface will be = 2πr × h = 2πrh

So the area of the cylinder will be: 2πr² + 2πrh, or

Total Surface Area of Cylinder = 2πr ( r + h )

Where r is the radius and h is the height of the cylinder (the distance between the two bases).

### The Volume of a Cylinder

Suppose if we have a cylinder of radius r and height h, then the volume will be,

V = πr²h

V is the amount of space occupied by the three-dimensional object. Let us see an example to find out the volume of a cylinder. As we know π = 3.14. So, let us find the volume of a cylinder that has the radius 3 cm and height 5 cm. Now,

V = πr²h

= π ( 3²) 5

= π ( 9 ) 5

= (3. 14) (45)

= 141.30 cm³

## Right and Oblique Cylinder

- Right Cylinder: When the two bases of the cylinder are over each other in exact position and the axis is at the right angle to the base, such cylinder is a right cylinder.

- Oblique Cylinder: When one of the bases of the cylinder is sideways and the axis is not a right angle to the base, then it is an oblique cylinder.

## Solved Examples For You

**Question 1. What length of a solid cylinder which is 2 cm in diameter must be taken to recast into a hollow cylinder of external diameter 20 cm, 0.25 cm thick and 15 cm long?**

- 54.06 cm
- 74.06 cm
- 34.06 cm
- 64.06 cm

**Answer:** B. The diameter of the solid cylinder = 2 cm or the radius = 1 cm; height h =?

V_{1} = πr²h = π(1)²h = πh

For the hollow cylinder, H = 15 cm; external diameter = 20 cm or external radius = 10 cm. Hence, internal diameter = 10-0.25 (thickness+ = 9.75 cm. Therefore,

V_{2} = π [ 10² – (9.75²) ] × 15 = 15π × 19.75 × 0.25

Also, V_{1} = V_{2}, which gives

h = 74.06 cm

**Question 2. If the lateral surface of the cylinder is 500 cm² and its height is 10 cm, then find the radius of its base.**

- 7.96 m
- or 7.96 cm
- 7.96 cm²
- 9.61 cm²

**Answer:** B. The lateral surface area is A =2πrh. The curved surface area is A = 500 cm² and its height is 10 cm, hence

A =2πrh

500 = 2 × 3.14 × r × 10

500 = 62.8r

r = 500/62.8

= 7.96

Therefore the radius of the cylinder is 7.96 cm

**Question 3: How to find the volume of a cylinder?**

**Answer:** For finding the volume of the cylinder, firstly, find the area of the base (which is a circle) by using the equation \( \pi r^{2}\) where r is the radius o the circular base. After that, multiply the area of the base by the height of the cylinder to know the volume of the base.

**Question 4: How many gallons are there in a cylinder?**

**Answer:** There is no fixed volume of how many gallons or liters a cylinder can hold. It depends on the size (radius and height) of the cylinder the bigger the cylinder the more gallons it can store.

**Question 5: State the formula of the total surface area of a cylinder?**

**Answer:**The total surface area of cylinder means that the outer surface of the cylinder plus the bottom and top surface of the cylinder. So, the general formula of the total surface area of a cylinder is \( 2 \pi rh + 2 \pi r^{2} \).

**Question 6: What is the total surface area of the hollow cylinder?**

**Answer:** The total surface area of a hollow cylinder is \(2 \pi r (r_{1} + r_{2}) (r_{2} – r_{1} + h)\). Here \(r_{1}\) is the inner radius \(r_{2}\) is the outer radius and h is the height.

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