# Cylinder Shape

By the end of this lesson, you will be able to

- define a cylinder.
- describe the base, height and axis of a cylinder.
- find the lateral surface area of a cylinder.
- derive the total surface area of a cylinder.
- introduce cylinder formula for area and volume.
- solve the problems involving surface area and volume of a cylinder.

## Cylinder

A cylinder is a solid object whose bases are two parallel congruent circles and body is a closed curved surface formed by the set of all points such that their distant from a fixed line segment is constant.

A cylinder showing its base, axis and curved surface.

The constant distant means the distant of all points from a fixed line is equal. The fixed line segment is formed joining the two centers of the base circles. This line segment is called the axis of the cylinder. And the length of the axis of a cylinder is called the height of the cylinder. So, a cylinder in 3-dimensions is a round closed shape with two congruent circled-bases.

## Base of cylinder

The top and bottom part of a cylinder are two parallel circles of equal area is called the base of the cylinder.

## Axis of cylinder

The line segment joining two centers of base circles is called the axis of cylinder. The length of axis is equal to the height of the cylinder.

## Height of cylinder

The perpendicular distance between two bases is called the height of a cylinder. Consequently, the length of axis is equal to the height because it is the perpendicular distance between two bases. Height is a very important part of a cylinder to evaluate the surface area and volume.

## Radius of cylinder

The top and bottom part of a cylinder are two parallel circles of equal area is called the base of the cylinder.

## Cylinder formula

A set of formula that are used to solve mathematical problems involving area and volume of cylinder is given below.

### Cylinder Example

## Area of cylinder

The area of cylinder means the total area of a cylinder. So, it is important to know how many parts there are in cylinder. A cylinder is formed with two base surfaces and a curved surface. Therefore, the sum of two base area and curved surface is called the total surface area of a cylinder.

### Area of base of cylinder

Top surface and bottom surface are well-known as the base surface of a cylinder. These two surfaces are circles that are congruent to each other. Since it is a circle, it has a center and a radius.

### Area of base of cylinder formula

Let the radius of base is r. Again, as it is a circle, its area is πr^{2} square unit. There are two base of same area in a cylinder.

Let the sum of two base area A_{base}. Then

A_{base} = πr^{2} + πr^{2}

∴ A_{base} = 2πr^{2}

If the radius of base of a cylinder is r and the sum of two base area is A_{base},

A_{base} = 2πr^{2}

### Curved surface area of cylinder

Curved surface area of a cylinder means the area of its round part. More precisely, the area of round surface that forms a pipe shape. Firstly, detach two circled-base surface from top and bottom. Then the cylinder will be a pipe shape whose top and bottom are empty.

Secondly, if we measure around the pipe with a string, its length is equal to the circumference of the base of cylinder and its value is equal to 2πr.

Thirdly, cut the pipe along to its length or height in direction, it produces a rectangle whose length is 2πr and width or breadth is equal to h, the height of the cylinder.

A cylinder is showing its radius and height.

### Curved surface area of cylinder formula

∴ Curved surface area of cylinder = area of the rectangle above

or, Curved surface area of cylinder = (length × breadth) square unit

or, Curved surface area of cylinder = (2πr × h) square unit

∴ Curved surface area of cylinder = (2πrh) square unit

If the radius of base of a cylinder is r, height is h and curved surface area is A_{curved},

A_{curved} = 2πrh square unit

## Area of cylinder formula

∴ Total surface area of cylinder = base area of cylinder + curved surface area of cylinder

Thus, if the total surface area of cylinder A, base area of cylinder A_{base} and curved surface area of cylinder A_{curved}

A = A_{base} + A_{curved}

বা, A = (2πr^{2}+2πrh) square unit

∴ A = 2πr(r+h) square unit

If the radius of the base of cylinder r, height is h and total surface area is A,

A = 2πr(r+h) square unit

## Volume of cylinder

For a regular solid object, the total surface area is equal to the area of base times the height. Cylinder is a regular solid object. So, if we multiply the area of the base of the cylinder by the height, we get the total surface area of a cylinder.

## Volume of cylinder formula

Let the radius of base and height of a cylinder be r and h respectively.

So, the area of base of cylinder = π r^{2} square unit

∴ Volume of cylinder = (area of base × height) cubic unit

Therefore, if the area of base A_{base}, height h and volume V,

V = A_{base} × h

∴ V = πr^{2}h cubic unit

If the radius r, height h and volume of a cylinder V,

V = πr^{2}h cubic unit